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Oi a) ba i 


A COURSE IN MATHEMATICAL ANALYSIS 


DIFFERENTIAL EQUATIONS 


BEING PART IL OF VOLUME II 


BY 


EDOUARD GOURSAT 


PROFESSOR OF MATHEMATICS, THE UNIVERSITY OF PARIS 


TRANSLATED BY 


EARLE RAYMOND HEDRICK 


PROFESSOR OF MATHEMATICS, THE UNIVERSITY OF MISSOURI 
AND 
OTTO DUNKEL 


ASSISTANT PROFESSOR OF MATHEMATICS, WASHINGTON UNIVERSITY 


GINN AND COMPANY 


BOSTON - NEW YORK + CHICAGO + LONDON 
ATLANTA + DALLAS + COLUMBUS + SAN FRANCISCO 


COPYRIGHT, 1917, BY 
EARLE RAyMonp HeprRIcK AND OTTO DUNKEL 


/ 


ALL RIGHTS RESERVED | 
PRINTED IN THE UNITED STATES OF AMERICA 


325.10 


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GINN AND COMPANY: PRO- | 
PRIETORS + BOSTON : U.S.A, 


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exp 4  # REMOTE STORAGE 


PREFACE 


The present volume consists of the second half of the second 
volume of the French edition of Goursat’s “Cours d’Analyse 
Mathématique.” As was stated in the preface to the first half 
of this volume, it seemed best, for purposes of American schools, 
to issue these two parts separately, and this was done with the 
approval of Professor Goursat. 

It is hoped that the present volume, which is entitled ‘ Differen- 
tial Equations,” will prove serviceable in American universities 


for courses which bear that name. 
E. R. HEDRICK 


OTTO DUNKEL 


iii 


CONTENTS 


PAGE 

CHAPTER I. ELEMENTARY METHODS OF INTEGRATION . . 3 
J. FoRMATION OF DIFFERENTIAL EQUATIONS . 

eeiMinavlonl OL COnsval bored caterer 2 tas cs) otc ie ke bs.) eee ee 

II. Equations oF THE First ORDER . 6 

2. Separation of the variables . 6 

3. Homogeneous equations . 8 

4. Linear equations ‘ 9 

DAP EULIOUL LE Sue UAbLOL Betree <7) hate Nadal e. pet hrer ts orca pajh tot) ou) Oe 

PehacoDUs req uaAmOIvar ality). vibes eh pkey poker mere le Mae ad) LL 

PeLVICCOteS ec UAUlOle Siete tel pein ge peek ok tea oo eo Le 

Bed ati Ola, DOUSOLVeECALOL 4) nekie Wa Mage Se fan yal oe ede ae et oy LA 

Dm LAC raAn TO StU MatIOMem a. | Uw 10 RG. he NR Abe os) 

10. Clairaut’s equation . . Aa Sb ey! 

11. Integration of the equations F G = “S 0, P ° = “0 i ee 

12. Integrating factors . . ae Lay ey). ese 

13. Application to conformal FonPeleniationt eehteet, ER Hoe see 

14. Euler’s equation . . AER ral ee ye es) 

15. A method deduced from Abel’s a neorer ey See Ee Oe! 

PG eS HOU oebeOLenis she) hg eee see Pest nn | Maas Man ao 

eMac Ul MSs Mn ee NM ese! alge ge el Maula Eas Wie Oe 

Pe PRUOUATIONS PORE ELIGHERVORDER ©...) aul vas bs OO 

Lom intesration of the equation @°y/dx" = f(z) 7.2. 4. ss 80 

PO my aliOUS Cases Of CEPresslony «) fap st. Get tee te pe OO 

BO MAD DUCALIONS 042° PewMolerma nile Merely ii wal@ siren de a oc TOS 

1 TETRIS ATS ISIE NE Sa ey Wr i er Ge aces We a 

Cir bint. BXISTENCE THEOREMS) (5. 3.27. 6 « oc. 945 

POR MILUSH ON LIMIT Gey). ho NI les a Oe eras 

21. Introduction. . . 45 

22. Existence of the iitepeale of ¢ a Em ae of differential datationé 45 

Zo, eyetems of linear equations)... 9.55 8 Us ee a 6 00 

Pa4eT el ditrerential equations. (yo. ree Be ee OL 


Vv 


vi CONTENTS 


PAGE 
25. Application of the method of the calculus of limits to partial 


differential equations . . Me 
26. The general integral of a ayetont of differential eiuatine . ee 


IL Tue Meruop oF SvuccesstvE APPROXIMATIONS. THE 


CatcHy-Lrpsourrz, MaTaoDil "520-2 a <n) ee 

27. Successive approximations 9.20 o- ae Ae he ee 
28. .Theease of linear equations’; 4 .¥ jx kip ig 2 eee 
29, Extension to-analytic functions <0 0." 7). 5) ee 
$0.° The Cauchy-Lipsenitz method” “49.0 3 a 
Til, Erest Inrecrarts. Muurppriers '. 9: 2.) 3 See 
$L..\ First ‘integrals: ae A, re oe 
32° Multiplierd ’ 2020 ny ye) ales toh acme Rann ie ae 
33. Invariant integrals 3. % os) ) sy aiyisi | Gel glee ola oh pik eee 


IV. INFINITESIMAL TRANSFORMATIONS . ....... 86 


34. One-parameter groups . . : os | MAPS i ae 
35. Application to differential eS teions MRE eS rsd are) 
36. Infinitesimal transformations.) sl. 4s" 94 95. ee 
FOXBBCESES. 8 Pe eas eee ey Ren egy he ne a en 
CHAPTER III. LINEAR DIFFERENTIAL EQUATIONS .. . 100 
I. GENERAL PROPERTIES. FUNDAMENTAL SysTEMS .. . 100 
37. Singular points of a linear differential equation . . . . 100 
38. Fundamental systems (5 0.5 2. 0. Wi, 0. 
39. The general linear equation . . . eT Oc. si 
40. Depression of the order of a linear equadon” oR Yep a 
41. Analogies with algebraic equations. . ...... . 118 
42. "The’adjomt equation 1" 39 6) 2 eee enn 
IJ. Tue Stupy or Some ParticutaR Equations .. <; 117 
43. Equations with constant coefficients . . . ... . . 117 
44. D’Alembert’s method” f.. 425 67.0.2) ee 
45. ‘Euler’s linear’equation. 9.0.3 3. ad Ae ee 
46. Laplace’s equation. 9). 0 bn ce sen ie 
III. Reeutar IntreerAts. Equations witH PERIODIC 
CoEFFIOIENTS 2! .0 5% a Fil, 9 Bee 
47. Permutation of the integrals around a critical point . . . 129 
48. Examination of the generalcase ......... Jl 


49, Formal expressions for the integrals . . . ... . . 1838 


50. 
51. 
52. 
53. 
54. 
55. 


CONTENTS 


Buchs. theorem, (+..| d2e Y eR a te 
Gauss’s equation 

Bessel’s equation . 

Picard’s equations 

Equations with periodic opafitetents 
Characteristic exponents 


IV. Systems or LINEAR EQUATIONS 


56. 
57. 
58. 
59. 
60. 
61. 
62. 


General properties 

Adjoint systems ny . 

Linear systems with tonseant anenicentst 

Reduction to a canonical form 

Jacobi’s equation . at SLA 

Systems with periodic poeineenes MAT dese oy fe nine 
Reducible systems 


EXERCISES . 


CHAPTER IV. NON-LINEAR DIFFERENTIAL EQUATIONS . 


I. ExcerTionAL INITIAL VALUES . 


63. 
64. 


The case where the derivative becomes infinite 
Case where the derivative is indeterminate 


II. A Strupy or Some EQuUATIONS OF THE First ORDER 


66. 
67. 
68. 


69. 
70. 


IIT. 


i. 
72. 
73. 
74. 


Singular points of integrals 

Functions defined by a differential ean ih y = iy co ) 

Single-valued functions deduced from the equation 
(y)"™ =k) - 

Existence of elliptic functions batter fare Buler’ s beer 

Equations of higher order . 


SINGULAR INTEGRALS. 


Singular integrals of an equation of the first order 
General comments 

Geometric interpretation ‘ : 
Singular integrals of systems of differential apaetiine 


EXERCISES . 


CHAPTER V. PARTIAL DIFFERENTIAL EQUATIONS OF THE 


FIRST ORDER 


I. LingAR EQUATIONS OF THE First ORDER 


75. General method : 
76. Geometric interpretation. 3): 4 « <' 8.6 « % 
77. Congruences of characteristic curves 


vil 
PAGE 
134 
140 
142 
1438 
146 
150 


152 


152 
156 
157 
161 
163 
164 
165 


167 


172 


172 
172 
173 
180 


‘180 
182 


187 
194 
196 


198 


198 
204 
207 
208 


212 


214 
214 


214 
218 
222 


vill CONTENTS 


PAGE 

Il. Toran DrrrerentTiat EQUATIONS ~~). 1.) «). % . Jeae2e 
70. Lhe equation dz = Adz 4 Bdy Wir a 0. a. oo. ee 
79. Mayer’s method .. . bee MM eet hh. 
80. The equation Pdx + Qdy + Rise = 0 Nia Oe a” cn te er ee 
81. The parenthesis (u, v) and the bracket [u, A ot a ae ee eee 


III. EeuUATIONS OF THE FIRST ORDER IN THREE VARIABLES 236 


82. Complete integrals . . . Po fa', tab he | os nn ne 
83. Lagrange and Charpit’s ideals cin, EE NiO RY toate a, ea 
84. Cauchy’s problem. . . PT Cr 
85. Characteristic curves. Caney: S method A elf 249 
86. The characteristic curves derived from a complete paral 259 
87. Extension of Canchy’s method. 92.3) s.. 0) 
LV. SIMULTANEOUS “EQUATIONS 0 0) =...) 0 eee) ee 
88; Linear homogeneous systems). ) i022 ss), es, one 
89. Complete systems. . . ; ; hee olen 
90. Generalization of the ieee of the Tes intenrals oc ee te - 
91. -Involutory systems; 4 > a OS en 
92. Jacobi’s method 51. .a) ajdt at 


V. GENERALITIES ON THE EQUATIONS OF HIGHER ORDER 278 


93. Elimination of arbitrary functions =. 2° = 2) ee 
94. General existence theorem... 2 2% .°7.Y 2S "See 
TOXERCISES- 3) -.. Geb his cl oe hy ee <i 


IN Dex ins Sore De AY a 


A COURSE IN 
MATHEMATICAL ANALYSIS 


VOLUME II. PART II 


DIFFERENTIAL EQUATIONS 


CHAPTER I 
ELEMENTARY METHODS OF INTEGRATION 


I. FORMATION OF DIFFERENTIAL EQUATIONS 


1. Elimination of constants. Let us consider a family of plane 
curves represented by the equation 


(1) F(a, y, ¢ yp Ca ***y Op) =O 
which depends upon n arbitrary constants. If we assign to these con- 
stants definite but arbitrarily chosen values, the successive derivatives 
of the function y of the variable x defined by the preceding equation 
are furnished by the relations 


OF OF 
OR Eh oes 
Or  . OF ae CRE ay 
(2) a te ta aa Y ere fe 
oF OF 
cee f+ — 7 — 
aah ess by 0. 


If we stop with the equation for calculating the derivative of the 
nth order, we shall have in all (n +1) relations between a, y, y’, y", 


-,y™ and the constants ¢,, ¢,,-+-,¢,. The elimination of these n 
aestnts leads in general to a single relation between a, y, y', ---, y™, 
(3) ® (x, Y) y'; y"; eb y™) = 0 


From the very way in which the equation (3) is derived it is clear that 
every function defined by the relation (1) satisfies this equation (3), 
whatever may be the values assigned to the constants c;; hence 
we say that any such function is a particular integral of the differ- 
ential equation (3). The whole set of these particular integrals is 
the general integral of the same equation. Using geometric language 
for convenience, we shall also say that every curve represented by 
3 


4 ELEMENTARY METHODS OF INTEGRATION [I, §1 


the equation (1) is an integral curve of the equation (3), or that the 
equation (3) is the differential equation of the given family of 
curves (1). 

We see that the order of the differential equation is equal to the 
number of arbitrary constants upon which that family of curves de- 
pends. It is also clear that the reasoning does not at all prove that 
the equation (3) has no other integrals than those which are repre- 
sented by the equation (1). In fact, the equation (3) may have other 
integrals, as we shall see presently. 


The above statements do not apply to the exceptional cases in which the 
elimination of the n parameters c; between the (n + 1) relations (1) and (2) leads 
to several distinct relations between a, y, y’, y”,---, y™. We could in those 
cases find one relation not containing y™, so that the family of curves con- 
sidered would be the integral curves of a differential equation of an order less 
than n. This will occur if these curves depend in reality upon only n— p 
parameters (p>0). For example, the curves represented by the equation 
F[z, y, ¢(a, 6)] = 0 apparently depend upon two arbitrary parameters a and 0; 
in reality they depend upon only a single parameter c = ¢(a, 0). There is also 
another way in which the lowering of. the order of the differential equation may 
occur. For example, the curves represented by the equation y? = 2azry + bx? 
really depend upon the two independent parameters a and b, yet these curves 
always satisfy the equation y= zy’. This is because the preceding equation 
represents two straight lines through the origin, each of which is an integral 
curve of the equation y = zy’. 

Examples. The straight lines passing through a fixed point (a, 6) are repre- 
sented by the equation 


(4) y—b=C(e—a) 
and depend upon an arbitrary parameter C. The elimination of this parameter 


between the preceding relation and the relation y’ = C leads immediately to 
the differential equation of this system of straight lines : 


(5) y—b=y7'(x— a). 
Conversely, we can write equation (5) in the form 


orate 
y—b t£—a 


and therefore every integral of that equation satisfies the relation 
Log (y — b) = Log (a — a) + Log C, 


which is equivalent to the equation (4). 
The set of all straight lines in a plane, y = C,2 + C,, form a two-parameter 
family whose differential equation is y” = 0. The converse is self-evident. 
The circles in a plane 


(6) e+y2+2Ar+2By+C=0 


I, § 1] FORMATION OF DIFFERENTIAL EQUATIONS 9) 


form a three-parameter family; the corresponding differential equation must 
therefore be of the third order. Differentiating the preceding relation three 
times, we find 


(7) etyy+At+BY=0, 1t+y?+ yy’ + By’ =), 
By y” + yy” + By” — 0. 
The elimination of B between the last two equations leads to the desired equation 

(8) y” (1 Le y?) pull 8 yy a 0. 

The only plane curves satisfying this relation are circles and straight lines. 
We see first of all that any straight line is an integral curve, for the equation 
is satisfied if we have y” = 0 and therefore y’” = 0. Now let us suppose that 
y’ #0; then we can write the equation (8) in the form 


4/7 


Wares Sy 
af > 1 + y? 


from which we derive 
S 
Log y” = 5 Log (1+ y”) + LogC,, 


where C, is a constant different from zero. This result may be written in the form 


A 


si K Soe a 
(+9)? 
A second integration gives 
Tw = C,7+ C,, 
or 
C124 + C2 


} V1i—(C,2 + C,)? 
integrating once more, there results finally 
Cyy + 0; =— V1—(C,2 + C,)?, 


which is the equation of a circle. 

The differential equation of all conics may be found easily by the following 
method, which is due to Halphen. If the conic has no asymptote parallel to 
the y-axis, its equation solved with respect to y is of the form 


y=me+nt+ VAx? +2 Be + C. 
After two differentiations we find 
oe AC— B 
~ (Aa? 42 Be + 0) 


47 


or 
(y’”)~ 3 = (AC — B?)- 4 (Aa? + 2 Br + C), 


so that (y’)—8/2 is a trinomial of the second degree in . Hence, to eliminate 
the three constants A, B, C three differentiations are sufficient, and the desired 
differential equation can be written in the abridged form 


a’ aay Lb 
qz3 LY) ]=9. 


6 ELEMENTARY METHODS OF INTEGRATION [I, §1 


Carrying out the differentiations, we obtain the equation 
(9) 40 y/"8 we 45 yy” yiv + 9 y yv — 0. 


The differential equation of parabolas may be found by the same method. 
We have, in fact, for a parabola A = 0, and (y’’)—24 is a binomial of the first 
degree. The differential equation is, therefore, in an abridged form, 

nie ee 
a Ly”) #] =9, 


or, after carrying out the indicated differentiations, 


(10) by” — 8yyiv = 0. 


1B EQUATION S OF THE FIRST ORDER 


Every differential equation of the nth order, formed by the elimi- 
nation of the constants, has an infinite number of integrals that 
depend upon n arbitrary parameters. But it 1s by no means evident 
that a differential equation given a priori has any integrals. This 
involves a fundamental question to which we shall return in the 
following chapter. We shall first consider some simple types of 
differential equations of the first order whose integration can be 
effected by quadratures. The existence of the integrals will be 
established by the very method by which we obtain them. If this 
order of procedure seems subject to criticism from the point of 
view of pure logic, we may at least observe that it conforms to the 
historical development of the subject. 


2. Separation of the variables. The simplest type of differential 
equation is the equation already studied, 


(11) “4 = f(@), 


where f(a) is a continuous function if the variable « is real, or an 
analytic function if we regard the independent variable x as com- 
plex. We have seen that that equation has an infinite number of 
integrals which can be represented by the relation 


y= J f@)ae+ C, 


where the lower limit’x, is considered as fixed, and where C denotes 
an arbitrary constant. The equation 


(12) oY = bly) 


I, § 2] EQUATIONS OF THE FIRST ORDER 7 


reduces to the preceding by considering y as the independent vari- 
able and # as the unknown function. The equation may then be 
written in the form dx/dy = a a and consequently 


», ? ot 


In general, when a differential equation is solved with respect to 
the derivative of the unknown function, it is often convenient to 
write it in the differential notation, 

(13) P(a, y)du + Q(x, y)dy = 0. 

This form does not commit us in any way as to the choice of the 
independent variable, which may be either a or y. If we wish to 
substitute for « and y new variables « and v, we need only replace 
x, y, dx, dy in the equation (13) by their corresponding expressions 
in terms of wu, v, du, dv. Let us also notice that we may, without 
changing the integrals of the equation (13), multiply or divide both 
its terms by the same function of # and y, u(a, y), provided that we 
take account of the solutions of the equation (a, y) = 0 which may 
be made to appear or may be suppressed by the operation. The two 
cases which we have just treated are particular cases under a more 
general method, called the separation of variables. If a differential 
equation of the first order is of the form 


(14) Xdx + Ydy =0, 


where X and Y depend only upon x and y respectively, we say that 
the variables are separated. The equation is then integrable by quad- 
ratures, for if we put 


x y 
v=f Xde+ f Ydy, 
%% % 


the equation can be written in the form dU = 0, and the ‘general 
integral is represented by the relation U = C. 
The equation 


(15) XY,dx +X,Ydy = 0, 


where X and X, depend only upon a, and where Y and Y, depend 
only upon y, can be reduced to the preceding form by dividing the 
two terms by X,Y,. It should be noticed that in this example the 
solutions of the ao equations, X,=0, Y, = 0, are suppressed. Indeed, 
it is clear that if y=bisa ea of fhe equation Y, = 0, y= 6 is an 
integral of the proposed equation, while in general it will not be 
included in the general integral of the new equation. 


8 ELEMENTARY METHODS OF INTEGRATION [I, §3 


3. Homogeneous equations. A differential equation of the first order 
is said to be homogeneous if it can be written in the form 


(6) ie) 


where the right-hand side is a homogeneous function of degree zero. 
It can be reduced to an integrable form by putting y = ux, where 
the new variables are 2 and wu. This substitution gives 


dx da’ 


and the equation (16) becomes 


du 
x +u=f(u). 


We can now separate the variables by writing the equation in the 
form 


Q 


De du 
fu)—u 


and the general integral is obtained by one quadrature in the form 


x 


du 
(17) x= Cer IM—*, 


We have only to replace in it w by y/x in order to obtain the equation 
of the integral curves. 

The general equation of that family of curves is of the form 
x =Cdo(y/x), where C is an arbitrary constant. These curves are 
all similar to any one of them, with the origin as center of simili- 
tude, the ratio of similitude being alone variable; for we can derive 
the preceding equation from the equation « = ¢(y/x) by replacing 
x and y in it by #/C and y/C respectively. Conversely, given a 
family of curves similar te each other with respect to the origin, the 
corresponding differential equation of the first order is homogeneous. 
We can verify this by actual calculation, but the result is evident 
a priori, for the tangents to the different curves of that family at 
the points of intersection with a straight line through the origin 
must be parallel, and therefore the slope of the tangent y' depends 
only on the ratio y/x. 

We can reduce to the homogeneous form any equation of the type 


dy _ (octets) 
(18) ae = a'x + bly +c! d 


I, § 4] EQUATIONS OF THE FIRST ORDER 9 


where a, 0, ¢, a', b', c' are any constants, except that d and b!' are not 
both zero. In order that this equation be of the desired form, it is 
sufficient that e=c!'= 0. Now, if we put 


x=X-+ a, y¥=Y+8, 


where X and Y are the new variables and where @ and B are any 
two constants, the given equation becomes 


=r a ge) 
aX * \aiXthY¥ + ae +B + c/’ 


and this new equation will be homogeneous if 
aa+bB+c=0, va+b'B+c'=0. 


These two conditions determine a and £ if ab'—a’‘b is not zero. 
In the particular case in which ab'— a'b=0, suppose b # 0; we 
shall have a'x + b'y = k(ax + by), where k is a constant which has 
a finite value. Putting ax + by = u, the equation takes the form 

Tdu_ a ( ute ) 


bdx 0b 


in which the variables are separated. 


4. Linear equations. A linear differential equation of the first 
order is of the form 


dy | 
(19) K+ Xy +X, =0, 


where X and X, are functions of x If X,=0, we can write this 
equation in the form 


(20) . +Xdx = 0, 
and the general integral is obtained by one quadrature in the form 


(21). yeeclat 


In order to integrate the complete equation (19), where X, is 
supposed different from zero, we shall try to satisfy that equation 
by taking for y an expression of the form (21), considering C’ no 
longer as a constant but as an unknown function of «. This 
amounts to making the change of variable y = Yz, where 2 is the 
new function to be determined and Y any one of the integrals 


10 ELEMENTARY METHODS OF INTEGRATION  [1,§4 


of the equation (20). After this substitution, the equation (19), 
by virtue of the relation (20) which Y satisfies, takes the form 


dz 
¥—-+X,=0, 
which is integrable by one quadrature. We derive from it 
Zs dx + C, 


where C is an arbitrary constant. The general. integral of the 
equation (19) is therefore obtainable by two successive quadratures. 
Replacing Y by its value, we can again write it in the form 


(22) es a iy ees “a), 


where the lower limits in the two integrals are chosen at pleasure. 

The general integral is an integral linear function of the constant 
of integration of the form y =Cf (x) + $(x), where f(x) and (x) 
are definite functions of x. This property characterizes the linear 
equation, for if we eliminate the constant C between the precens 
equation and the equation 


y' =CH@)+ $'@), 
we are evidently led to a relation that is linear in y and y/. 
This result may be stated in another way. Let y,, y,, y, be three 
particular integrals of the linear equation, corresponding to the 


values C,, C,, C, of the constant C’; the elimination of the two func- 
tions f(x) and ¢(x) between the three relations, 


Y=CLOE)+O@), W=CFO)+ EO), Y= CSO) + O(~@), 


_ leads to the relation (y, — y,)/(y, — y,) =(C; — €,)/(C, — C,), which 
shows that the ratio (y,— y,)/(y,—y,) is constant for any three 
particular integrals of a linear equation. If we know two particular 
integrals y,, y, of a linear equation, we can then write down imme- 
diately the general integral in the form 


Yas 

Y.-~ Nn 

It is also to be noticed that if we know a single particular inte- — 

gral y,, the general integral can be obtained by a single quadrature ; 

in fact, putting y = y, + wu, we are led to the equation du/da+ Xu=0, 
which is identical with the equation (20). 


= const. 


I, § 6] EQUATIONS OF THE FIRST ORDER 11 


5. Bernoulli’s equation. Bernoulli’s equation 
dy : a 
(23) ch + xy + Ky" =0, 


where the exponent m may be any number different from zero and 
from unity, can be reduced to a linear equation by the substitution 
z#=y'—". For then we can write the preceding equation, after 
dividing all its terms by y”, in the form 


eed = : 


We can reduce to the preceding type any equation of the form 


(24) (4) dx + (4) dy + ka"(ady — ydx)= 0, 


where & and m are any two numbers whatever. For if we put 
y = ux, the equation obtained can be written as follows: 


d 
[ p(w) + up (u) | oe tur) kare * 2 a=. 0, 
and, putting 2 = 2- “t+, we are led to a linear equation. 


6. Jacobi’s equation. Let us consider the equation 


(25) { (a+ ax + a’y) (xdy — ydz) 
—(b+ ba + b’y)dy+(c+cx+c’y)dz = 0, 


where a, a’, a”, 0, b’, b”, ¢, c’, c” are any constant coefficients. Ifa@=b=c=0, 
the equation comes under type (24), for we have only to divide by a’ + a’y to 
reduce it to this type. In order to reduce the general case to this particular 
case, let us putt = X + a, y= Y + B, where X and Y are two new variables 
and where a and £ are two constants. Thus we obtain a new equation of the 
‘same form, which can be written as follows: 


(aX + a”Y)(XdY — YdX) 
(25’) —(B4+0UX4+0°Y-(A4+a@X+a”"VY)a—AXIAY 
+[C+ eX 4+ce°Y—-(A+aX+4+a’Y)B—AY]|dX =0, 
where © 
A=a+a@”a+a’p, B=b+0 ad's, C=c+cat+c'p. 


This equation (25’) will be of the type (24) if we have Aa— B=0, AB— C=0. 
We are then led to determine the constants a, B by these two conditions, which 
may be written in a more symmetric form by introducing an auxiliary un- 
known 2: 


AS 0y PLB Anais 0,4) Ce ne = 0: 


12 ELEMENTARY METHODS OF INTEGRATION [I, § 6 


The elimination of the unknowns a, B leads to an auxiliary equation of the 
third degree for the determination of \: 
a—nr a’ a” 
b bo —X Doe ee 0. 
c c’ ce’ —X 


The integration of Jacobi’s equation depends, then, first of all on the solution 
of this equation of the third degree, as will be seen by other methods a little 
later. 


7. Riccati’s equation. Riccati’s equation 
ad 3 
(26) ; a. + Xy'+ Xy + X,=0, 


where X, X,, X, are functions of x, cannot in general be integrated 
by quadratures. The integrals of this equation, when the coefficients 
are unrestricted, form new transcendental functions, whose proper- 
ties we shall study. But this equation is related to the matter which 
we are discussing on account of the following property: Jf we 
know a particular integral, we can find the general integral by two 
quadratures. 

Let y, be a particular integral. The change of variable y = y, + 2 
leads to an equation of the same form which does not contain any 
term independent of z, since x = 0 must be an integral; that equa- 
tion is, in fact, 


d 
(27) Fe + (X, + 2Xy,)2 +Xe = 0, 


and we have only to put «w =1/z in order to transform it into a 
linear equation. This proves the proposition just stated. 

From this result, several important consequences follow. The 
general integral of the linear equation in w is of the form (§ 4) 


u=Cf(x)+ o(x); 


hence the general integral of the Riccati equation is of the form 


ere cmmianas) Sean aE), 

VAT OD TE@. FOTO) 

We see that it is a rational function of the first degree in the constant 
of integration. Conversely, every differential equation of the first 
order which has this property is a Riccati equation. For, let f(x), 
(x), f(x), $,(#@) be any four functions of x; all the functions y 
represented by the expression (28), where C is an arbitrary con- 
stant, are integrals of an equation of the first order, which is easily 


I, § 7] EQUATIONS OF THE FIRST ORDER 13 


obtained by solving the equation (28) for C and then taking the 


derivative. This gives 
C= $y a. yp : 


A eA 


and the corresponding differential equation is 


(uf — fi) (1 — bY’ — ¥b')— (bi — YD) OF + Uf! — SF) = 9, 
which is of precisely the form (26). 
Let ¥, Yq, Ys) Y, be four particular integrals corresponding to the 
values C,, C,, C,, C, of the constant C. By the theory of the anhar- 
monic ratio we have the relation 


Yarra Ne MU oWO ey ah, he sine Th 

Yn Yn UY Vg CoN. rae 
which is easily verified also by direct calculation, and which proves 
that the anharmonic ratio of any four particular integrals of Riccati’s 
equation is constant. 

This theorem enables us to find without any quadrature the gen- 
eral integral of a Riccati equation when we know three of its partic- 
ular integrals y,, y,, y,. Every other integral y must be such that 
the anharmonic ratio (y — y,)/(y — y¥,) +(Y3 — ¥,)/(Ys — Yp) 18 con- 
stant. The general integral is then obtained by equating this ratio to 
an arbitrary constant. It is clear that y will be a rational function 
of the first degree in this constant, which proves that the preceding 
property belongs only to the Riccati equations. 

Let us observe that if we know only two particular integrals, y, 
and y,, we can complete the integration by one quadrature; for, 
after the first transformation y = y, + 2, the equation obtained in z 
has the integral y, — y,. The linear equation in wu has therefore the 
known particular integral 1/(y,—y,). The general integral of the 
equation in « will then be found by a single quadrature.* 


Application. Let us consider a family of circles in a plane, which depends 
upon one variable parameter. Let (a, b) be the codrdinates of the center of the 
variable circle and let R be its radius (the axes being rectangular). We shall 


* The properties of Riccati’s equation established in the text can be derived also 
by observing that the equation is not changed in form by any general linear trans- 
formation y= (/z+)/(7,2+ %1), where f, /;, ¢, ¢, are functions of x. If we know 
one, two, or three integrals of the equation (26), we can always choose the linear 
transformation in such a way that, in the transformed equation in z, one, two, or 
three of the coefficients of the polynomial of the second degree in z will be zero. A 
linear equation may be regarded as a Riccati equation which is satisfied by the 
particular integral y=, that is, such that the equation obtained by putting y=1/2z 
has the solution z= 0, 


14 ELEMENTARY METHODS OF INTEGRATION [1, §7 


suppose that a, b, R are known functions of a variable parameter a. Let us try 
to find the curves which cut each of these circles at a known angle V, which 
may be constant or a given function of a. The codrdinates of any point M of 
the circle C with the center (a, b) and the radius R can be represented by the 


equations 
z=a+Rkeoos6, y=—b+Rsin8, 


where @ is the angle which the radius terminating at the point M makes with 
the direction Ox. The problem reduces to the determination of the angle 6 as a 
function of the parameter a, so that the curve described by the point M cuts the 
circle C at the angle V. The differential equation of the problem is therefore 


ee Ba 8: 
x 
cthn V = ———,, 


1+ tan 0 dy 
dx 
which becomes, after replacing dx and dy by their values and reducing, 
dé 7 Ue ed “A 7 . Cgertics 
eae b’ cos 6 — a’ sin 6 — ctn V (R’ + a’ cos@ + b’ sin 6) = 0, 
a 


where a’, b’, R’ are the derivatives of a, b, R with respect to a. Taking for the 
new unknown ¢ = tan (0/2), we obtain the Riccati equation 


(29) 2R + b(1— #2) — 2a’t— ctn V[R(1 + @) + a’ (1— 2) + 20] =0. 
a : 


It will suffice, then, to know a single trajectory in order to obtain all the others 
by two quadratures. 

Let us consider the particular case of orthogonal trajectories ; the angle V is 
then a right angle, and the cotangent is zero. If we also suppose that the circles 
considered have their centers on a straight line, we know a priori two particular 
integrals of the equation (29), for the line of the centers is an orthogonal tra- 
jectory and meets each circle in two points. It is easily shown that the inte- 
eration requires only one quadrature, for if we take the z-axis for the line of 
centers, the equation (29) reduces to R (dt/dz) — a’t = 0. 


8. Equations not solved for y’. In the different cases which we 
have just examined the equation was supposed to be soived with 
respect to y'. Let us now consider the general equation of the first 
order F(x, y, y')= 0. Let S be the surface represented by the equa- 
tion F(x, y, 2) = 0, obtained by replacing y' by z. To every integral 
y = f(a) of the proposed equation there corresponds a curve I’, rep- 
resented by the relations 


(I) y=f@), 2=f'@); 


which lies entirely on the surface S, since we have 


Fle, f(@), f'(@)] = 0. 


I, § 8] EQUATIONS OF THE FIRST ORDER 15 


But this curve I is not any curve on the surface S; along this curve, 
in fact, y and z are functions of # satisfying the relation dy—zdx=0, 
and that relation preserves the same form if we take any independ- 
ent variable in place of «. 

Conversely, let I be a curve lying on the surface S; the coérdi- 
nates x, y, 2 of a point of that curve are functions of a variable a. 
If these three functions, « = $,(@), y = $,(@), # = $,(@), satisfy the 
relation dy = zdx, we can deduce from them an integral of the given 
equation ; for the first two relations, « = ¢,(@), y = $,(@), represent 
a plane curve C. Let y= f(x) be the equation of that curve, suppos- ' 
ing it solved for y. Along the entire curve I we have z = f"(ax), and 
consequently F[a, f(x), f(x) ]= 0; the curve C is therefore an. inte- 
gral curve. There would be an exception only in case the curve C 
were to reduce to a point, and the curve I to a straight line parallel 
to Oz. The two problems are then equivalent: to integrate the given 
equation F(x, y, y')= 0 or to find the curves of the surface S§ for 


which we have 
dy — zdx = 0. 


This being the case, let us suppose that we can express the coér- 
dinates of a point x, y, z of the surface S explicitly as functions of 
two variable parameters wu, v: 


ee hss Y= (U,V), ae wu, v). 


Every curve I of the surface S is obtained by establishing a certain 
relation between uw and v, and, in order that that curve shall define 
an integral, it is necessary and sufficient that we have dy = zdz, or 


op 
du 


du + > e 


dv = W(u, (ZF du oa ta av): 
We have thus a differential equation dv/du = m(u, v), solved with 
respect to dv/du. It is clear that the preceding discussion applies 
also to equations which can be solved for y’. 

This transformation is immediate for the equations solved for one 
of the variables x or y. For example, let the equation be 


(50) y=f(@, y')3 


we can here take for the variable parameters x and y' = p. The sur- 
face S is then represented by the equations 


c=, k2=p, y=Sf(x,p); 


16 ELEMENTARY METHODS OF INTEGRATION [1,§ 8 


and the relation dy = zdx becomes 


_ of , of dp 
4) Papen 


This result could have been obtained directly by differentiating the 
relation (30) and replacing y' by p. Let p = (a, C) be the general 
integral of the equation (31); to deduce from it the general integral 
of the equation (30), it will only be necessary to replace y’ in the 
equation (30) by $(a, C). 


9. Lagrange’s equation. Let us consider in particular an equation 
linear in the two variables x and y: 


(32) y = xoly') + Hy’). 
Differentiating the two sides, and denoting y' by p, we obtain the 
equation 
dp dp 
Ros ! —- ! Saad 
Pea OUP te Dr ate YE is 


If we consider p as the independent variable, and a as the unknown 
function, that equation, which can be written in the form 


[o(p) —P] a + x'(p) + ¥'(p) = 9, 


is linear and is integrable by two quadratures. Having obtained x 
as a function of p, by putting that value of x in the expression __ 


y = xh(p) + WP); 
we Shall have the codrdinates x and y expressed as functions of the 
parameter p and of an arbitrary constant.* 


We can readily discuss the general appearance of the family of integral 
curves by observing that x and y are polynomials of the first degree in the 
arbitrary constant C: 


(33) a = CF(p) + ®(p), y = CF,(p) + ®,(p). 


But the functions F(p), F,(p), ®(p), &,(p) are not arbitrary functions, since 
the parameter p represents the slope dy/dx of the tangent. On this account 
we must have Fi(p) = pF’(p), ®i(p) = p®(p). Let Ty, I’, be two particular 
integrals corresponding to the values C= 0, C=1 of the constant: 


Ly = P(D), rp f%1 =F (p) + &(P), 
"LY = 2,(P), "= Fi(p) + 2,(p). 


* The equation (32) can also be reduced to a linear equation by means of Legendre’s 
transformation (I, § 62, 2d ed.; § 36, 1st ed.). 

A homogeneous equation of the form y= x¢(y’), not solved for y’, may be consid- 
ered as a particular case of Lagrange’s equation and integrated in the same way. 


I, § 10] EQUATIONS OF THE FIRST ORDER 17 


The equations (83), which represent any integral T, may be written also in 


the form 
a = C(&, — %) + %, 
Y = CY; — Yo) + Yo 
At the points M)(%, Yo), M,(x,, y;), M(a, y) of the curves Ty, T,, T, which 
correspond to the same value of p, the tangents to these curves are parallel. 
Moreover, we derive from the preceding expressions 
Y-Y t—-% £4C 
Y¥—Y, e—2, C—1’ 


which proves that the three points M, M), M, are on a straight line and that 
the ratio MM,/MM, is constant. We have then the following geometric 
construction: Given the two curves Ty, T,, we join the points M,, M, of these 
two curves where the tangents are parallel, and we take on the straight line joining 
these points the point M such that the ratio MM,/MM, will be equal to a given 
constant K. If the points M,, M, describe the curves 1), T',, the point M describes 
an integral curve I, and we obtain the general integral by varying the constant K. 


10. Clairaut’s equation. A remarkable particular case of Lagrange’s 
equation had been treated previously by Clairaut; every equation of 
the form 


(34) y =ay'+ f(y’) 


is called a Clairaut equation. Following the general method, we 
‘differentiate the two sides and put p =y’'; this leads to the equation 


(35) [etre Zao. 


This equation is satisfied by putting dp/da =0; whence p=C. The 
general integral of Clairaut’s equation is, then, 


(36) U == Ot ae CY: 
This equation represents a family of straight lines, and it is readily 
seen that they are really integral curves. But the equation (35) is 
also satisfied by causing the first factor x + f'(p) to vanish, From 
this it follows that there exists a new integral of the equation (34), 
which is represented by the two equations 


e+ f'(p)=0, y=petf(p). 
Now the elimination of p between these two equations would lead 
precisely to the envelope of the straight lines represented by the 
equation (36). Hence Clairaut’s equation has also as an integral 
curve the envelope of the straight lines which represent the general 
integral. Since we cannot obtain this integral by giving a particular 
value to the constant C, we say that it is a singular integral. 


18 ELEMENTARY METHODS OF INTEGRATION _ [I,§10 


We are led to Clairaut’s equation when we undertake to determine a plane 
curve by a property of its tangents in which the point of contact does not enter. 
In fact, let y = f(x) be the equation of the desired curve ; then the equation of 
the tangent is Y= y’X + y — zy’, and we are led to a relation between y’ and 
y — xy’, that is, to Clairaut’s equation. It is clear that in this case it is the 
singular integral which gives the real solution of the problem. 

Let us propose, for example, to find a curve such that the product of the dis- 
tances from two fixed points F, F’ to any one of its tangents is always equal to a 
constant b?. Let 2c be the distance FF’, let the middle point of the segment 
FF’ be taken for the origin, and let the straight line FF’ be the z-axis. This 
leads to the differential equation 

(y— ay’)? — c2y/2 = 2 (1+ y’?) 
if we suppose that the two points F, F’ lie on the same side of the tangent. This 
equation reduces to the form y = zy’ +Vb? + aty? ; hence the general integral 
represents the family of straight lines 


y=Ceive?+a?C?,  a=t? + c%. 
The singular integral curve, the ils of these straight lines, is the ellipse 


ida 
which is the true solution of the problem. 


11. Integration of the equations F(x, y!)=0, F(y, y!)=0. The 
equations which contain only one of the variables x or y are inte-. 
grable by a quadrature, provided that we can solve the equation for 

' ($2). If the equation is algebraic, y is an Abelian integral or 
the inverse function of an Abelian integral. Whenever the relation 
is of deficiency zero or deficiency one, we can express x and y as 
functions of a variable parameter, either rationally or by means of 
the classic transcendentals. Let us consider, first, equations of the 
type F(y, y')= 0, of deficiency zero; we can express y and y! as 
rational functions of a parameter u, y= f(u), y' =f,(w), and the 
condition dy = y'dx gives us f'(u)du = f,(w)dx. Then the variables 
x and y are given by the expressions 

me LOCO ay 

(37) y F(X); Ge ae 
in terms of the variable parameter vu. The same procedure is applica- 
ble to the equations F'(y, y')= 0 if the relation is of deficiency one ; 
but we must take for f(w) and f,(w) elliptic functions, and «# and y are 
expressed in terms of the transcendentals p, ¢, o (Part I, § 75). 

We can proceed similarly with the equations F(a, y')= 0 if the 
relation is of deficiency zero or one; besides, they reduce to the pre- 
ceding form by interchanging a and y. 


I, § 12] EQUATIONS OF THE FIRST ORDER 19 


Example 1. The equation y? (y’ — 1) = (2 — y’)? is of deficiency zero. Putting 
2—y = yu, we derive from it y =1+4 u2,y=1/u—u. The relation dy = ydz 
becomes here dx =— du/u?. We have, then, x = 1/u + C, and the general inte- 
gral of the given equation is y =z —C—1/(«—(C). 

Example 2, The equation y* — 3y’2 — 9y* — 127? = 0 represents, if we regard 
y and y’ in it as the coérdinates of a point, a unicursal quartic having three 
double points (y = 0, y’ = 0), (y=+V— 2/3, y’ =2). We can, in fact, write 
the preceding equation 

(y’ — 2)? (y’ + 1) = By? + 2). 
Putting first y’=u?—1, we have 3y2?=(u+1)?(u— 2); if we then put 
u— 2 = 37, we obtain finally the following expressions for y and y’ as func- 
tions of the parameter ¢ : 
y=3( + 1), y = 38(14+ t) (14+ 30%). 
The relation dy = y’dx reduces here to (1 + t#)dz = dt; we derive from it 
t=tan(¢+C), 
and the general integral of the given equation is therefore 
y = 8tan(x + C) + 3tan?(z + C). 


Example 3. Let R(y) be a polynomial of the third or of the fourth degree, 
prime to its derivative ; let us consider the differential equation 
(38) YA al). 


We have seen in § 78, Part I, that we can satisfy this equation of deficiency 
one by putting y=f(u), y =f’(u), where f(u) is an elliptic function of the 
second order. The condition dy = y’dx becomes du = dz ; the general integral 
of the equation (88) is therefore an elliptic function y = f(x + C). 

If the polynomial F (y) is of lower degree than the third, or if the polyno- 
mial, although of the third or of the fourth degree, is not prime to its derivative, 
the relation (38) is of deficiency zero. We can express y and 7’ by rational func- 
tions of a parameter u, and, by applying the preceding method, we easily show 
that the general integral is a rational function of « or a rational function of e*, 


12. Integrating factors. The method of integration by the separa- 
tion of the variables was generalized by Euler. The reasoning of § 2 
apples really to every equation of the first order 

(39) P(2, y) da + Q(x, y)dy = 0, 
where the coefficients P and Q contain both a» and y, provided that 
we have 0P/éy = @Q/éx. This condition is necessary and sufficient 
in order that Pdx + Qdy shall be the total differential of a function 
U(x, y), and the function U (a, y) is obtained by quadratures, as we 
have seen (I, § 151). The equation (39) is then identical with the 
equation dU = 0, and the most general solution is given by a rela- 
tion of the form U(x, y)=C between x and y. The equation (39) is 
therefore integrable by quadratures whenever the coefficients P and 
Q satisfy the condition ¢P/éy = 0Q/éx, 


20 ELEMENTARY METHODS OF INTEGRATION _ [I1,§12 


In. order that the preceding method may be applied, it is not 
necessary that we have ¢P/éy = 0Q/éx; it suffices to know an inte- 
grating factor, that is, a factor w(a, y) such that the product 


u(w, y) [Pax + Qdy] 


satisfies the integrability condition @(mP)/dy = (MQ) /éx, or, after 
developing, 
é é GPEC 
(40) pe aH + ul 2) = 0 


The investigation of the integrating factors is thus reduced to the 
integration of the preceding equation, which is a partial differential 
equation of the first order. It seems that in proceeding in this way 
we have made the integration of equation (39) depend on an appar- 
ently more difficult problem, but it is to be noticed that it suffices 
to know one particular solution of the equation (40) in order to apply 
the method, and in many cases we can find a particular integral of 
the equation (40) by more or less direct processes. Let us see, for 
example, in what case the equation (39) has an integrating factor 
depending only on x. If we suppose 0u/cy = 0, the equation (40) 
becomes 


and the expression [0P/éy — 6Q/éx]/Q must be independent of ¥; 
if it is, we obtain an integrating factor mw by a quadrature. Let us 
suppose in addition that Q@=1; then @P/éy must be a function X 
of the variable x, and the equation (39) is a linear equation, 


(39') dy + (Xy + X,)dx = 0, 


where X and X, denote functions of x alone. In this case, the equa- 
tion (40) is satisfied by 
f- Xax 
a Er re Ng 


and it 1s easy to show that if we multiply the equation (39') by this 
factor, we have on the left an exact differential 


ela t (dy is Xyda + X,dx) = ay ek ae +f Xe a 0. 
The calculations which have to be made for the integration are 
exactly the same as in the first method (§ 4). 

We shall show farther on that the equation (40) has an infinite 
number of integrals under very general conditions, which are always 
satisfied in the cases in which we are interested. If we know one 


I, § 12] EQUATIONS OF THE FIRST ORDER a1 


integrating factor w,, we can obtain all others in the following way: 
Putting w = p,v, the equation (40) becomes 


(40’) a do 


Now we know one function satisfying this relation: it is the func- 
tion, U(x, y), whose total differential is u,(P dx + Qdy), since the 
partial derivatives ¢U/cx, CU /¢y are equal to w,P and to 4,Q. We 
have, then, also (év/0y) (@U /éx) — (év/ex) (CU /ey) = 0, which proves 
that v is of the form ¢(U) and that the general expression for the 
integrating factors is w = u,p(U), where ¢ is an arbitrary function 
of U. It is easy to show that w is really an integrating factor, for 


from the identity 
M,(P dx + Qdy)= dU 


we derive, by multiplying by #(U), 


H,b(U)[P (a, y)dx + Q(a, y) dy|= $(U) dU, 
and the right-hand side is the exact differential of the function 


R(U)= f o(wyav. 


We deduce from this an interesting consequence: if w, and pm, are 
two integrating factors, the ratio w,/m, is a function of U. If this 
quotient w,/u, is not constant, the general integral of the differential 
equation can then be written in the form pw,/u, = constant. 

The preceding theorem is sometimes helpful in finding an inte- 
grating factor. Let us consider the differential equation 


(41) Pdz + Qdy + P,dz + Q,dy = 0, 


where P, P,, Q, Q, are functions of a, y, and let us suppose that we 
know how to find an integrating factor for each of the expressions 
Pda + Qdy, P,dz + Q,dy. The general expression for the integrat- 
ing factors of Pdx + Qdy is wp(U), where pw is the known factor, U 
a function of # and y which we obtain by quadratures, and ¢ an 
arbitrary function. Similarly, the general expression for the inte- 
grating factors of P,dx + Q,dy is m,w(U,), where mw, and U, are 
definite functions and y an arbitrary function. If we can choose the 
functions ¢ and yw in such a way that we have 


pp (U)= 4,4 (U,); 


we shall have an integrating factor for the given equation (41). 


22 ELEMENTARY METHODS OF INTEGRATION _ [1,§12 


Let us take, for example, the equation 
axdy + bydz + amy(axdy + Bydzx) = 0, 


where a, b, a, 8 are constants. Every integrating factor of axdy + bydz is of 
the form ¢(x?y*)/zry, and, similarly, every integrating factor of the second 
part is of the form y (x8 y2)/am+1lyr+1, In order to have a common integrating 
factor, it will suffice to find two exponents, p and g, such that we have 


gmyn (x? y%)P = (x8 y%)4, 
which leads to the conditions , 
pa—qatn=0, pb—qB+m=0. 


These conditions are compatible if a8 — ba is not zero, and determine an inte- 
grating factor of the form z%y¥, Multiplying by this integrating factor, the 
equation takes the form v?—1dv + v{—-!dv, = 0, where we have put v = 2 y4, 
v, = x? y; and this equation is immediately integrable. 

In the particular case where a8 — ba = 0, we obtain from it a/a = B/b=k, 
and the equation can be written in the form (ardy + bydz) (1+ kr™y”) = 0. 


Note. If we know the general integral of a differential equation of the first 
order, it is quite easy to obtain an integrating factor. For let f(z, y) = C be 
the general integral of the equation (39). The differential equation of the curves 
represented by that relation is also (@f/0x) dx + (af/cy)dy = 0; in order that it 
be identical with the equation (39), we must have 


OF tos: 
ae _ ay 
PGA 


and the common value of the two preceding ratios is evidently an integrating 
factor for Pdr + Qdy. Every other integrating factor is equal to this one 
multiplied by an arbitrary function of f(x, y). 


13. Application to conformal representation. The theory of integrat- 
ing factors finds an important application in the problem of conformal 
representation. Let 


ds? = Edw? + 2 Fdudv + Gdv’ 
be a quadratic form in du, dv whose coefficients EH, F, G are analytic 


functions of u and v such that EG — F? is not zero. We can also 
write ds? in the form 


ds? = (adu + bdv) (a,du + b,dv), 
where a, 6, a,, 6, are also analytic functions of w and v. According 
to a result which will be rigorously proved later, each of the expres- 
sions adu + bdv, a,du + b,dv has an infinite number of integrating 


factors, which are themselves analytic functions. If pw, mw, are two 
such factors, we have the identities 


u(adu + bdv)= dU, L,(a,du + b,dv)= du,, 


I, § 14] EQUATIONS OF THE FIRST ORDER 238 


and therefore 
pp,ds* = dUaU,; 
whence, substituting 


U=X+Y7i, U,=X—Y, bh, =F? 


we obtain 
Edw + 2 Fdudv + Gdv? = (dX? + dY”). 


Every analytic surface can therefore be represented on a plane 
conformly ; that is, without alteration of the angles between pairs of 
curves. If the surface is real, we may suppose that the real points 
of the surface correspond to real values of the variables u, v; the 
coefficients EF, F, Gare real, while a and a, are conjugate imaginaries, 
as also b and 6,. We can also take for » and yw, and therefore for U 
and U,, conjugate imaginaries, so that to real values of u, v corre- 
spond real values of X and of Y. To real points of the surface 
correspond therefore real points of the plane. 

Since it is possible to represent every analytic surface on a plane 
conformly, we conclude that any analytic surface can be represented 
conformly on any other analytic surface. 


14. Euler’s equation. A great many devices have been invented for 
the integration of differential equations of special forms. A cele- 
brated example, due to Euler and now known by his name, is the 
equation 


dix dy 
Co Decals Aaa 


where X and Y are two polynomials of the fourth degree in x and y 
respectively, having the same coefficients : 


ae 4 3 2 
A=a,x + 4,x° + a,0° + a,4 + a, 


Y=ay+ayt+ay +a y+ 4,. 


The variables being separated, we obtain the general integral of 
equation (42) by two quadratures, which introduce two transcen- 
dental functions depending respectively upon « and y. Euler’s fun- 
damental discovery, which was the starting point of the theory of 
elliptic functions, consisted in showing that that relation between 
the variables x and y which in appearance is transcendental is in 
reality algebraic. 

Let us first consider the case where X is a polynomial of the sec- 
ond degree, not a perfect square. A linear substitution enables us to 


24 ELEMENTARY METHODS OF INTEGRATION _ [1,§ 14 


bring it to the form X =A (a —1), and in this particular case the 
equation (42) becomes 
dx dy 
43 SS 
at Vi-a V1i—2 
Clearing of fractions, we can write this in the form 


V1 — ytde + V1 — a2dy = d(x V1—- y+ yV1—2?) 


0. 


which shows that we have identically 
d(aV1— 2 + y V1 — x*) 
re Gai dx d 
= V/ 1—2)(1— At, CGS ae 2 ) 
[ ( )¢ y’) y] REE aa Vi-# 
The expression V (1 — x) (1 — y’) — xy is therefore an integrating 


factor for the equation (43), and the general integral is given by the 
relation | ; 


(44) aV1l—-y+yV1—-v=0C, 
or by the relation 
(45) Vid—2)A—¥)—ay=C, 


since the equation (43) has the two integrating factors, 1 and the 
expression on the left-hand side of (45). It is also very easy to 
verify that the two expressions (44) and (45) are equivalent by 
means of the identity 


(aVi- P+yVi-—a) + Py (ee) ay | =1. 


Rationalizing the expression (45), we can write the general integral 
of the equation (43) in the form 

(46) et+y+t2clry+c"%—1=0, 
where C' denotes an arbitrary constant, and this equation represents 
the conics tangent to the four straight limes x=+1,y=21. 

By a bold induction Euler was led to a more general formula of 
the same kind, which corresponds to the case where X is any poly- 
nomial of the third or of the fourth degree (Institutiones caleuli 
integralis, Vol. I, chaps. v, vi). 

Let F(x, y) be a polynomial of the second degree in each of the 
variables « and y and symmetrical with respect to these two variables : 


(47) F(a, ¥) =A cy +A, xy (x + ¥) 
+A, (a? +Y)+A,xy +A,(@ + y) +A, 


I, § 14] EQUATIONS OF THE FIRST ORDER 25 


This polynomial depends upon six arbitrary coefficients 4,, A,, A,, 
A,, A,, A,, and the relation F(#, y)=0 can be written in two 
equivalent forms: 

(48) He eames Ate 8 = 0, 

F(@, y)=M,2*+N,2 +P,=0, 


where M, N, P are three polynomials of the second degree in a: 
M=A,x'?+A,a aren N=A,2'+A,2 + Ay; PA op As Ar. 
and where I, N,, P, are the polynomials obtained by replacing a by y 


in M, N, P. From the relation F(x, v) = 0 we derive Fda + F/dy =0, 
or, after replacing F, and F;, by their values, 


(49) (2M,2 + N,)dx+(2My+N)dy=0. 
We derive, moreover, from the relations (48), 
2My+N=t+VN°—4MP, 2Mx2+N,=+VN?—4MP, 
and the preceding equation (49) may be written in the form 


dx by dy E 
VN? —4MP) VN?—4M,P, 


This relation will be identical with the given equation (42) if we 
have N?—4MP =X, which necessarily carries with it the other 
equality N{j—4M,P,=Y. Now, since M, N, P are of the second 
degree, VN? — 4 WP is of the fourth degree, and the preceding condi- 
tion is an identity between two polynomials of the fourth degree, 
which requires only five conditions. Since we have six coefficients 
A, at our disposal, we see that one of these coefficients will remain 
arbitrary. There are therefore an infinite number of polynomials 
F(x, y) of the form (47), depending upon an FOUR constant C 
and such that the relation 


(51) F(a, y)=0 


between the variables x and y, leads to the relation (42). Hence the re- 
lation (51) represents the general integral of the proposed equation. 


(50) 


The actual determination of the polynomial F(a, y) requires a calculation by 
equating coefficients which can be simplified by means of a geometric repre- 
sentation due to Jacobi. Let us consider, in order to take the general case, a 
polynomial of the fourth degree R(t) prime to its derivative, and let ¢,, t,, t,t, 
be the roots of R(t) = 0. On the other hand, let = be any conic the codrdinates 
of any point of which are rational] functions of the second degree of the varia- 
ble parameter ¢, so that to a point (#, y) corresponds a single value of ¢; let us 


26 ELEMENTARY METHODS OF INTEGRATION _ [1,§14 


call m,,m,, M3, m, the points of 2 which correspond to the values ¢,, f,, ts, t, 
of the parameter. Finally, let 2’ be a second conic passing through the four 
points m,, m,, ms, m,. Every straight line tangent to =’ meets = in two points 
M and M’; if t and v’ are the corresponding values of the parameter, the rela- 
tion between t and ¢’ is the one desired. It is evident, in fact, that that relation 
is symmetric in ¢ and ¢’, and that it is of the second degree in each of the varia- 
bles, for through a point M’ we can draw two tangents to 2’, and so to each 
value of t’ correspond only two values of ¢. 
Let 


(52) F(t, ’) =0 


be that relation. We can derive from it, as we have just seen, a relation 
between the differentials dt, dt’, of the form 


ae dt’ 
VP@) VEO) 
where P(t) is a polynomial of the fourth degree. This polynomial P(t) is iden- 
tical except for a constant factor with R(t); for, according to the preceding 
method for obtaining the polynomial P (t) from F(t, t’) = 0, the roots of P(t) = 0 
are the values of ¢ for which the two values of ¢’ coincide. Now the geometric 
significance of the relation (52) shows immediately that this can only occur if 
the two tangents from M to 2’ coincide; that is, if the point M is one of the 


points m,,m,, m3, m4. We are thus led to the following method, which requires 
only rational calculations, for obtaining the general integral of the equation 


(53) ax), 


dt a 
VR() VR) 


where R(t) = a)t* + a2 + at? + a,f + a,. This equation differs only in nota- 
tion from the proposed equation (42). We begin by forming the general 
equation of the conics 2’ passing through the four points m,, m,, m,, m, of Z; 
that equation is of the form f(z, y) + C¢(a, y)= 0, where C is an arbitrary con- 
stant. We then write the condition that the straight line joining the two points 
M and M’ of 2, which correspond to the values ¢, t’ of the parameter, shall be 
tangent to 2’. The resulting relation, which contains the arbitrary constant C, 
represents the general integral of Euler’s equation. 

To carry out the calculations, let us take for = the parabola y? = a, and let 
us put 2 = t?, y=t. The conic 2’ given by the equation 


(54) 


(55) Ag? + A’y? +2 B’ cy +2B’x+2By+ A” =0 


cuts Z in four points, given by the equation of the fourth degree in t which is 
obtained by replacing x by t? and y by t. In order that that equation shall be 
identical with R (¢) = 0, it is sufficient that 


(56) A= Oy, A’ 412 B Sd, 2B = Oy, ate ete A een 
The coefficient B’ remaining arbitrary, we shall put B’ = C, which gives 


A’=a,— 20. 


I, § 14] EQUATIONS OF THE FIRST ORDER oT 


Let us recall now that the tangential equation of 2’, that is, the condition that 
the straight line az + By + y = 0 shall be tangent to that conic, is given by the 


equation ld eA eee 
Dene Mite fe 8 oe 0 
Bie By AREY luli 

fa Ve 

The straight line joining the two points (é?, t) and (t, t’) of = has for its 


equation e—(t+t)y+v=0. 


(57) 


We can therefore take 
(8 ie B=—(t+?), nyt bb 
Substituting the values obtained for A, B, A’, B’, A”, B’, a, B, y in the con- 


dition (57), and replacing ¢ and ¢’ by z and y respectively, we arrive at the gen- 
eral integral of Euler’s equation in the following form, which is due to Stieltjes: 


ay 
ao ny C 1 
ay a 
— O@—2C0- = —(aty 
(58) 2s 2g Ah ek 3 
C 3 ay Ly 


1 —(@+y) wy 0 


This equation represents a family of curves of the fourth degree, having two 
double points at infinity on Oz and Oy respectively. The equation being of the 
second degree with respect to the constant C, through each point of the plane 
there pass two curves of the family, as we might have foreseen, since the given 
differential equation gives two equal values, but with opposite signs, for the 
derivative y’ at each point. These two values of y’ become equal only if the 
point (x, y) belongs to the curve XY = 0, which is composed of four straight 
lines D,, D,, D,, D, parallel to the axis Oy, and of four straight lines A,, A,, A;, Ay 
parallel to the axis Oz. Let us write Euler’s equation in the rational form 
Y dx? — X dy? = 0, and let us take a point M (a, y) on one of these straight lines, 
A, for example, not belonging to any one of the D lines. For the codrdinates 
of the point M we have Y = 0, X~0, and Euler’s equation gives for y’ a double 
value, y’ = 0. Hence the straight line A, itself is an integral curve through M. 
But it can be verified that the curves represented by the equation (58) have as 
their envelope the set of eight straight lines given by the equation XY = 0. 
Hence there is a new integral curve tangent to the first one at M. Thus the 
eight straight lines D;, A; are singular integral curves, for they are not included 
among the curves represented by the general integral. 


Note. .We have supposed, in order to arrive at the equation (58), that the 
polynomial R (x) was one of the fourth degree and prime to its derivative ; but 
it is clear that the result can be verified directly without the hypothesis that 
R (zx) is prime to its derivative. We could, for example, form the differential 
equation of the curves represented by the equation (58) by applying the general 
method of § 1, and the equation obtained would necessarily be identical with 
Euler’s equation, whatever may be the values of the coefficients a), a, dj, M3, A, 
since we reach this result when the coefficients do not satisfy any particular 
relation. The equation (58) therefore applies to all cases. 


28 ELEMENTARY METHODS OF INTEGRATION  [1,§ 15 


15. A method deduced from Abel’s theorem. We can also very easily deduce the 
general integral of Euler’s equation from Abel’s theorem. Let us now denote 
by R(x) a polynomial of the third or of the fourth degree, prime to its deriva- 
tive, and let us consider the curve C which has for its equation y? = R(2). 
If a variable algebraic curve C’ meets the curve C in three variable points 
only, M,, M,, M,, we have shown (Part I, § 103) that the codrdinates (a,, v1)s 
(Ly5 Yo)s (4, Ys) a these three variable points satisfy the relation 
ry | ity, dy _ 
Viv URN ave 

If the variable curve C’ depends upon two variable parameters which we 
can select in such a way that two of the points of intersection, (7, y,), (2, Yo); 
can be brought to coincide with any two points of the curve C given in advance, 
the codrdinates of the third point of intersection, (#,, y,), are functions of the 
codrdinates (x1, 1; Lg, Y2) of the first two, and satisfy the relation (59). The equa- 
tion dz,/y, + dz,/y_= 0 is therefore equivalent to the equation dz,/y; = 0, whose 
general integral is 7, = constant. Now, since the points (x,, ¥;), (2, y_) are on the 
curve CO, we have yi = R(z,), y3 = R(z,), and the equation dz,/y, + dx,/y, = 0, 
which may be written in the form 


(59) 


Ut,» dz. 
60 ee ee 
te VE (t,) VR (a) 


is identical with Euler’s except in notation. In the expression which gives the 
general integral 


(61) Le F(L5 Ys Los Yo) = CODE. 


we should replace y, and y, by VA(x,) and VR(x,) respectively, the deter- 
minations of the two radicals being the same in the two expressions (60) 
and (61). We thus obtain for the general integral an expression containing 
radicals, while the result (58) is rational. But the irrational form is in certain 
cases the more advantageous. 

Let us carry out the calculations, supposing the polynomial R(x) reduced to 
the normal form of Legendre, R (x) = (1 — x) (1— kx”), where k? is different 
from zero and from unity. The parabola 0’, 


(62) y = ax? + be +1, 
meets the curve C represented by the equation y? = R(z) in the point (4 = 
= 1) and in three variable points whose abscissas 2,, x, 2, are roots of i 
Pr on 
(63) (a* — k*) x? + 2Qaba* + (0? + 2a4+k?41)r4+20=0, 
which is obtained by eliminating y and suppressing the factor z. 
We derive from this equation the relations 


2 ab 674+ 2a+k?4+1 
Lt, +t, +2; = eT y Lo + Lol, + £1, = [Ce 
26 


aS | 
k2 — q? 


? 


whence 


(64) Ly + Ly + Lz = AL, Voy. 


I, § 16] EQUATIONS OF THE FIRST ORDER 29 


The condition that the parabola ©’ passes through the two points (z,, y,), 
(2, Yo), enables us to determine a and 6. We have in particular 


Yi %_ — Yor 

: L, — Ly 
Substituting this value of a in the preceding expression, we obtain finally the 
expression for x, in terms of 2, ¥;, %, Ye: 


2 2 
T2Y1 — T%Ye 


The general integral of Euler’s equation, 


3 = 


dx, dx, 


(65) a a ee i) 
VR(z,) VE(@) — 
is therefore represented by the expression 
xo? — x2 
(66) fe 


t,VR (1) — @,V R (a2) 


16. Darboux’s theorems. Let us consider a differential equation of the form 
(67) — Ldy + Mdz + N (ady — ydz) = 0, 


where L, M, N are three polynomials in x, y of at most the mth degree, and 

where at least one of them is actually of the mth degree. In order that the 
relation u(x, y) = constant shall represent the general integral, it is necessary 
and sufficient that the equation (67) be identical with the equation 


Se dr + dy = 0, 
Ox 


which requires that we have 
(68) E+ M uc — N(x6 ot ey =) = 0. 
oy 


This condition assumes a more symmetric form if we replace x by x/z and y 
by y/z, where z is a fictitious variable which we shall always suppose equal to 
unity after the indicated operations have been performed. Then u(z, y) 
changes into a homogeneous function of degree zero, and we have 


ee) Let MS + Ne A (u) = 
Conversely, if we have obtained a homogeneous function of degree zero, 
u(x, y, Z), Which satisfies the relation (69), w(x, y, 1) = constant represents the 
general integral of the equation (67). 

Darboux* has shown that we could form a function u (a, y, z) satisfying 
these conditions if we knew a certain number of algebraic integrals of the 


* Sur les équations différentielles algébriques du premier ordre et du premier 
degré (Bulletin des Sciences mathématiques, 1878). 


30 ELEMENTARY METHODS OF INTEGRATION _ [1,§ 16 


equation (67). Suppose that the equation (67) has an algebraic integral defined 
by the relation f(x, y) = 0, where the polynomial f(z, y) is irreducible and of 
degree h. Repeating the previous work, we find that the relation 


traf Cs of of 
(70) Lares N(2Z + yZ) = 0 


must be a consequence of the equation f(z, y)=0. If we again replace x by 
x/z, and y by y/z, and then multiply by 2%, f(z, y) becomes a homogeneous 
function of x, y, z, of degree h, satisfying the relation 


oye oS = We, 


and the condition (70) becomes 
(71) CP) ad eM wil = wy, 


This condition is not satisfied identically, but by reason of the relation 
f(x, y, 2) =0. Since the last relation is irreducible by hypothesis, it is neces- 
sary that we have identically 


(72) A(f) = Kf, 


where K denotes a polynomial in x, y, z which is necessarily of degree m — 1, 
for if f is of degree h, A(f) is of degree m+ h—1. 

Let us now suppose that we have found p algebraic solutions of the equa- 
tion (67), defined by the p following equations : 


S,(2, y) = 90, Sg (&, y) = 9, PERS Sp (x, y) =0 


where f,, f.,:::, fp are irreducible polynomials of the degrees h,, hg,-++, Ap. 
This requires that we have p identities of the following form : 


(73) A(f)=Ayh, A(f,) = Kyh,, eats A (fp) = Kp hp, 
where the polynomials K,, K,,---, K, are all of degree m— 1. 

Let us observe that the symbolic operator A(f) has properties analogous to 
those of a derivative. In particular, we can apply to it the rule for the deriva- 
tive of a function of functions: if F(u, v, w) is any function of u, v, w, we have 


A(F) = o A(u) “i (0) + o A(w). 


. a a, 
Consequently, if we put u=f("1 fy... te where @,, @,°+*, @p are any con- 
stants, we have 


A(u) = Leh iNaiew Bi ‘ +. fer A(f,) + a, fe t cmeana Sp? (Sy) fees, 


or, by (78), 
A (u) = (a, K, + a, Ky + +++ + ap Kp)u. 


The function u(x, y, 2) is a homogeneous function of degree 
ah + Aghgt +++ + aphp. 
If we can dispose of the constants a, --+ @» in such a way that we have 


(74) eee +++ aphp = 9, 
Ri ee ee, 


I, § 16] EQUATIONS OF THE FIRST ORDER 31 


the equation u(x, y, 2) = constant will furnish the general integral of the given 
equation, by what we have established above. 

The equations (74) form a system of m(m+1)/2+1 homogeneous equa- 
tions in Qj, @,+++, @, since the polynomials K; of degree m— 1 contain 
m(m-+1)/2 terms. We shall surely be able to satisfy all these equations by 
values of a; not all zero, and therefore to complete the integration, whenever 
there are more unknowns than equations ; that is, whenever we have 


(75) explana), 


This is Darboux’s first theorem. If the equations (74) are not independent, 
we can find the solutions without requiring p to reach the preceding limit 
m(m-+1)/2+ 2. A large number of examples in which this is the case will 
be found in Darboux’s paper. 

If we know only p = m(m + 1)/2 + 1 particular algebraic integrals, we can, 
in general, dispose of the p constants a; in such a way as to satisfy the conditions 


(76) 1h Bm ih AG as wr kit dB te mee omy ’ 


Ah, + aghy + +++ + Aphyp =—m— 2, 


which are equivalent to a system of m(m + 1)/2 4+ 1 linear non-homogeneous 
equations. The function w thus obtained satisfies the two equations, 


ou CU Ou COLAO MeO: 
Lotus see rons 
Ou 
ota eee = + (m+ 2)u=0 


whence we derive, by rs éu/éz and replacing z by 1, 


ou oL oM . @eN 
LS+ MS —w[ mt user +y alt" (set oy + ae) =o 


But, since the function NV has been made homogeneous by substituting x/z for z 
and y/z for y, and then multiplying by 2”, we also have, after making z = 1, 


so that the preceding relation may be written also in the form 
(77) - (L — Ne) + - (M— Ny) 
x 


Soe et yea) a. 
y 


It is easily seen that this last condition expresses the fact that u is an integrat- 
ing factor for the equation (67), and we obtain thus Darboux’s second theorem : 


If m(m+ 1)/2 +1 particular algebraic integrals of the equation (67) are known, 
an integrating factor can be determined. 


The proof of this last theorem is not complete in the particular case where 
the determinant of the coefficients of the unknowns qj; in the m(m+1)/2+1 


32 ELEMENTARY METHODS OF INTEGRATION _ [1,§ 16 


equations deduced from the relations (76) turns out to be zero, But we can then 
satisfy the m(m + 1)/2 + 1 homogeneous equations, obtained by suppressing the 
right-hand sides, by values of the a; not all zero, and therefore obtain the 
general integral by the first theorem. 

Example. Let us consider in particular Jacobi’s equation (§ 6); the num- 
ber m is here equal to 1. Let us look first for the linear integrals of the form 
uz + vy + wz = 0. By the general method we must have identically 


u(bz + 0’ + b’y) + v(cz + ca + CY) 
+ w(az+ ae + a’y) = (ux + vy + we), 


where X is a constant factor. This leads to the three conditions 


ub + ve + w(a— rd») =), u(b’ — dr) + ve’ + wa’ = 0,7 
ub” + v(c” — dA) + wa” = 0, 
and, after eliminating u, v, w, we find again the equation in \ obtained by the 
first method (p. 12). 

Let us limit ourselves to the case in which the equation in X has three dis- 
tinct roots \,, 4,, 43. Each of these roots furnishes a linear integral, and we 
therefore have three linear functions, X, Y, Z, giving the three identities 

As(20) =I Xe AY NS, Nig A (Z) =A,Z. 


By the general theory we can deduce from them the general integral, since 
in this case m =1. For this purpose it is necessary to determine three numbers 
a, 8, y satisfying the relations 


at+B+ry=90, AD, + Bry + YA3 = 0. 
We may take a =, — dg, B = Ay — Ay, Y = Ay — Ag, and the general integral of 


Jacobi’s equation is therefore 


XA2— AsYAs—A1ZA1— Az = const. 


17. Applications. When we seek to determine a plane curve by a 
given relation F(x, y, m)=0 between the coordinates (a, y) of a 
point on the curve and the slope m of the tangent at this point, the 
curves desired are evidently obtained by the integration of the differ- 
ential equation of the first order F(a, y, y')=0, which we obtain 
from the given relation by replacing in it m by y'. If this equation 
is of the gth degree in y', there pass in general g such curves through 
each point of the plane, as will be proved farther on. Let us con- 
sider, for example, a family of curves C, represented by the equation 
@(x, y, a) = 0, depending upon an arbitrary parameter, and let us 
try to find their orthogonal trajectories, that is, the curves C’ which 
cut orthogonally in each of their points a curve C passing through 
the same point. Let m, m' be the slopes of the tangents to the two 
orthogonal curves C, C' passing through the same point (a, y). Then 
mand m' must Bpas ty the relation 1+ m'm=Q. On the other hand, 


I, § 17] EQUATIONS OF THE FIRST ORDER 33 


let F(a, y, y') = 0 be the differential equation of the given curves C. 
Then we have F(a, y, m) = 0, since m is the slope of the tangent to 
a curve C’ passing through the point (a, y). It follows that 


(2, Y; -=)=0 


Moreover, m! is also the slope of the tangent to a curve C’ passing 
through the point (#, y); hence the curve C’ satisfies the equation 


1 
(78) (2, Hele i) = 0, 


and we obtain the differential equation of the orthogonal trajectories 
of the curves C by replacing y' by —1/y' in the differential equation 
of the curves C. , 


In order to obtain the differential equation of the curves C, we must 
eliminate a between the two equations ® = 0, (¢&/dx) + (0@/dy) y'= 0. 
Therefore, in order to obtain the differential equation of the orthogonal 
trajectories, it will suffice to eliminate a between the two relations 
& = 0, (Cb/Cx) y' — (C&/ey) = 0. 

Let us take, for example, the conics represented by the equation 


y+32?—2ax=0, 


where a@ is a variable parameter. The application of the preceding 
method leads to the homogeneous differential equation 


(y — 32°) y' + 2ay =0, 


which becomes, after putting y = ux and separating the variables, 


x x: U u+ti u-1 
Solving this equation, we find 
ute CO(UR— 1) Or = CY, — 2). 


The orthogonal trajectories are therefore cubics with the origin as a 
double point. . 

Let us consider in a more general manner a surface S the coérdi- 
nates x, y, of any point of which are expressed as functions of 
two parameters w, v: 


w= f(u, Vv), y = o(u, v), 2=wW(U, v). 


34 ELEMENTARY METHODS OF INTEGRATION [I, § 17 


We derive from these expressions 


de=Z aus L ar, dy = 52 au + SS ao, dz = aut & ae, 
To every value of the ratio dv/du corresponds a tangent to the sur- 
face passing through the point (wu, v). If we wish to determine the 
curves of that surface such that the tangent to one of these curves in 
any point depends only on the position of that point on the surface, 
we are again led to integrate a differential equation of the first order : 


dv\ 
(79) | F (u, v, =) aH Ve 


Conversely, every equation of this form establishes a relation between 
a point of a curve lying on the surface S and the tangent at that point. 

Let us, for example, try to find the trajectories at a constant 
angle V to a family of given curves lying upon the surface. Given 
two curves, C’, C', passing through a point (wv, v) and cutting at an 
angle V, we have the general formula (II, Part I, § 20) 


Eduéu + F(dudv + dvdu) + Gdvbv 


80 Cos bE OF eee 
Cy, VEde + 2 Fdudv + Gdv? VE 8? + 2 FSudsv + G8v? 


where E, F, G have the usual meanings, where dw and dv denote 
the differentials relative to a displacement on C, and where du and $v 
denote the differentials relative to a displacement on C'. The curves 
C' being given, 6v/du is a known function of wu and v, dv/du = 7 (u, v). 
Replacing 6v/du by a (wu, v) in the preceding relation (80), the result- 
ing relation F(u, v, dv/dw) = 0 is the desired differential equation of 
the trajectories. 

Let us consider in particular the trajectories at a constant angle 
to the meridians of the surface of revolution, 


x = p Cos a, ¥Y=p sin W, i == F(p). 
We have here 


PR D0), E=1+/f"%p); F= 0, G =p’, du = 0; 


hence the equation (80) becomes 


en otf vV1i+f"(e)dp 
V[1+F%(p)]dp? + pdo” 
Solving for dw, we find 
12 
p 


whence can be obtained by a quadrature. 


I, § 18] EQUATIONS OF HIGHER ORDER 35 


III. EQUATIONS OF HIGHER ORDER 


18. Integration of the equation d"y/dx" = f(x). Given a differen- 
tial equation of the nth order, 

(81) ve Pe F(a, Y; y', y", i oea? ise), 
where y® = d'y/dz', this equation and those which are obtained from 
it by repeated differentiation enable us to express all the derivatives, 
beginning with y™, in terms of a, y, y', y",---, y®—». If, then, for a 
particular value x, of the independent variable we are given the cor- 
responding values y, yj, +--+, y§”» of the unknown function y and of 
its m —1 first derivatives, we can calculate the values of all the 
derivatives of y for the value «x, of x, and form a power series, 


eee 
(82) t@—a)y + FS yt +. 4 CaM yoy, 


whose value represents the integral in question, provided that inte- 
gral can be developed by Taylor’s series. Up to the time of Cauchy’s 
work the convergence of this series had been assumed without 
proof.* We shall see later that the series does converge under cer- 
tain conditions which will be stated precisely. We shall indicate 
here only some simple types of differential equations of the nth 
order whose integration can be reduced to quadratures or to the 
integration of an equation of lower order than n. 
The differential equation: 


(83) 


ay _ 
die) 


constitutes the simplest possible type of differential equation of the 
nth order. It can be integrated by means of » successive quadra- 
tures; for, indicating by #, any arbitrary constant, we have 


im =| F@)det Oy 


ea =f af f(a)da+C,(@—2,)+C, 


ys fae fa f 4 F(x) dx 


4 Cyt — Xo) aa C,(@ — a%)"~? 


(n—1)! (n — 2)! 


test Oya, 


* See, for example, the Traité by Lacroix, 


36 ELEMENTARY METHODS OF INTEGRATION _ [I,§18 


where C,,_1, C,_2,+++, C, are n arbitrary constants which are equal 
respectively to the values of the integral and of its first (n —1) 
derivatives for « = m,. 

We can replace the expression 


v= [def ae... f F(a) da, 


which contains n successive signs of integration, by an expression 
containing only a single quadrature, to be carried out on a function 
in which the variable x appears only as a parameter. It is easy to 
verify this fact, which will appear later as a special case of a general 
theory (§ 39). For if we put 


1 x 
(84) 1G = (w—1)! f (a = z+ f(2) dz, 
we obtain successively, by the application of known rules, 


dy. 1 , cea) 
E=aeDL Cro TOs -) Gara) Teme 


and, finally, d”Y,/dx" = f(x). The function Y, is therefore an inte- 
gral of the equation (83). Besides, the two functions Y and Y, vanish, 
as do also their first (n —1) derivatives, for «=«,. Their differ- 
ence, which is a polynomial of degree equal to nm — 1 at most, cannot 
be divisible by («—a,)" unless it is identically zero. We have 


therefore Y. = Y. 


1 


19. Various cases of depression. The most usual cases in which the 
order of the equation can be depressed are the following: 

1) The equation does not contain the unknown function. An equa- 
tion of the form 


Aa Ve hee n 
(85) F(a, SE, Tt, SY 20 (1=ksn) 
reduces immediately to one of order n — k by taking u = d*y/dz* as 
a new unknown function... If the auxiliary equation in w can be inte- 
grated, we shall then obtain y by quadratures, as has just been 
explained. | 

It sometimes happens that we can express # and u = d*y/daz* in 
terms of an auxiliary parameter ¢, 


k 


r=f(), Tt=9(, 


I, § 19] EQUATIONS OF HIGHER ORDER 37 


where the functions f and ¢ contain also the arbitrary constants 
introduced by the integration of the equation in uw. We can then 
express y in terms of ¢ also by quadratures. We have first 


dy*-» = $(t)dx = $( fat, 
whence we derive y*—». Continuing in this way, we calculate suc- 
cessively y*—®,..-, y' up to y. 
2) The equation does not contain the independent variable. Given 
an equation of the form 


dy d’y d"y 
(86) r(y, $4 4... SH) = 0, 
we can reduce it to the preceding form by taking y for the independ- 
ent variable and x for the unknown function. Then the new equa- 
tion does not contain x, and, taking dz/dy for the new unknown, we 
are led to an equation of order » —1. But we can carry out these 
two transformations simultaneously by taking y for the independent 
variable and p = dy/dzx for the dependent variable. This gives 


DUNE IDE ID CY aD 
dx? dx dy dx * dy’ 


dty d/ dp\_a( dp dy aap. 
dat? = al? al? 7)? »(Z) TP ay 


and so on. In general, d”y/dx” can be expressed in terms of p and 
of its first 7 — 1 derivatives with respect to y. The resulting differ- 
ential equation is of order n — 1. 

Let us suppose that we have integrated this auxilhary equation of 
order m — 1, and for the sake of generality let us suppose that y and 
p are expressed in terms of a variable parameter ¢, which may be one 
of the variables themselves. Then we shall have y= f(¢), p = $(4), 
where the functions f and @ depend also on arbitrary constants. 
From the relation dy = pdx we derive f'(t) dt = $(¢) da, so that a in 
turn is obtained by a quadrature, 


t'@) 
o(f) 


This method is especially useful for the equation of the second order, 


x= 


Fiy, y', y')=9 
which is thus reduced to an equation of the first order, 


d 
F(np. pit) =0. 


38 ELEMENTARY METHODS OF INTEGRATION _ [1;$19 


Let p = $(y, C) be the general integral of this equation of the first 
order. From the relation dy/dx = ¢(y, C) we obtain x by a quadrature, 


fa _ dy 
seats b(y, C) 


If the general integral of the equation in p is solved for y and 
appears in the form y= /f(p, C), we have, in the same way, 


I'(p) dp = p dx 
Aa F'(p) ap | 
x= x, =p ie 


The coérdinates of a point of an integral curve are thus expressed 
in terms of an auxiliary variable p which represents the slope of the 


and therefore 


tangent to the curve. 
3) The equation is homogeneous in y, y', y", +++, y. Tf the degree 
of homogeneity is m, the equation is of the form 


foe ty, (n) 
(87) mr ( aration ot 


and we see that, if y, is a particular integral, Ay, is also an integral 
for any value of the constant A. The order of this equation is 
lowered by unity by putting 

y et od" dx, 


This substitution gives 
y' = wes, y" = (ul + uv’) ote, nate 


and, in general, y is equal to the product of “ah "@ and a polynomial 
in wu, u', w"',.-+,u’—-». Substituting these values in the given equa- 
tion, we obtain an equation of order n — 1. 

4) The equation is homogeneous in x, y, dx, dy, d’y,--+-,d”y. In 
this case the equation is not changed by substituting Ca for x, and 
Cy for y, where C' is any constant. Let us now take a new dependent 
variable u = y/x and a new independent variable ¢ = Loga. The 
new differential equation does not change if we replace ¢ by ¢+ Log C, 
leaving w unchanged; hence it does not contain explicitly the vari- 
able ¢. This is readily verified, for it is easy to see that the given 
equation must be of the form 


q 
r(¥, y', ay", arty", toe ry ) ==) 
ab 


{, § 20] EQUATIONS OF HIGHER ORDER 39 


If we put y = ua, we have, as a general expression, 


y — ruP + pu?—?, 


and the quantities y', xy", 2?y'", --- are expressible in terms of wu, ww’, 


xul',+.+, xu™, so that the transformed equation takes the form 
D Gee, cortel vontacl! Meera), =O; 


If we now put « = e’, we have successively for the products aw, 
xu'',.-- certain functions of du/dt, d?u/dt’, ..-, and we are led to 
an equation which does not contain the variable ¢.* 


Note. In the various cases of reduction which precede, it may 
happen that we can obtain certain integrals of the auxiliary equation 
without being able to determine the general integral. The preced. 
ing methods are still applicable and enable us to obtain by quadra. 
tures integrals of the given equation containing less than n arbitrary 
constants. 


20. Applications. 1) Equations of the form 7” = f(y) come under the preced- 
ing types. We can integrate them directly without any transformation, for if we 
multiply the two sides by 2 y’, we deduce from the result, by a first integration, 


y 
y? = C+ fi 2f(y) dy = Fy) + C, 


and we have next, by a quadrature, 


7; 


cs | ee 
VF (y) + 6 
Let us consider, for example, the equation 
y= ay? + yy? + ayy + ag, 
where one at least of the coefficients a), a, is not zero. Multiplying the two 
sides by 27’ and integrating, we find 


shila 2 
y? stk +3 uy? + ayy" + 2agy + C. 


The general integral of this new equation is an elliptic function (§ 11), which 
may in special cases reduce to simply periodic functions, or even rational func- 
tions, if the constant C has been so chosen that the polynomial on the right has 
a factor in common with its derivative. 


* We may proceed in another way by taking wu and v= xw’ for the variables. This 
gives dv/dx=wu’ + xu”, and therefore x2u” = (dv/du) u’x — xu’, or 
dv 


22u”’ =v ——v. 
du 


Continuing in this way, we are led to a differential equation of order (n—1) in 
uw and v. 


40 ELEMENTARY METHODS OF INTEGRATION [I, § 20 


2) It may happen that we can apply successively several of the methods of 
reduction to the same equation. Let us take, for example, the equation of the 
fourth order 57”? — 3y”yiv = 0. If we first put y” = u, we derive from it an 
equation of the second order, 5u’* — 3 uu” = 0, which is homogeneous in u, u’, u”. 
Let us put 


ares Ale dz | 

the equation becomes 3 v’ = 2 v?, or v’/v? = 2/3, from which we obtain 
Late Dean 
ht 2 ie a. 


where a is an arbitrary constant. Hence we have 


Ui Yo = (Ge a)~ 2, 

y =—2b(e+ a)~2 +¢, 

y=—4d(e+ a)? + ce + d, 
where b, c, d are three new constants. We find, therefore, that the general 
integral represents a system of parabolas (§ 1). 

3) Let it be required to determine the plane curves whose radii of curvature 
are proportional to the portion of the normal included between the foot M and 
the point of intersection N of that normal with a fixed straight line. Taking 
the fixed straight line for the z-axis, the differential equation of the problem is 

(88) T+ y7 + wy” =9, 
where the coefficient « is equal to the ratio of the radius of curvature to the 
length MN, preceded with the sign + or —, according as the direction from M 
to the center of curvature coincides with the direction MN or with the opposite 
direction. In order to integrate this differential equation (88), let us put 
y’ =p; it becomes dp 


1+ p* + nyp—— = 0, 
dy 


which can be written in the form 
ek fae a aE 
Cis Bene 
from which we derive, by a first integration, 
y=C(L +p?) 


where C is an arbitrary constant. The relation dy = pdz gives us next 


b] 


2 
’ 


Be 
=e] 
— pCp(1+p?) 2 dp=pdz, 
or 


Me 
reel 
@ = ay — nC f (1+ p?) 2 dp. 


Let us put p = tana; all the curves obtained by varying C and z, result 
from a translation and an expansion about the origin of the curve [I represented 
by the equations 


a 
(T) L=— af cos ada, Y = COBF a, 
0 


It is easy to get an idea of the form of the curve from these equations, what- 
ever may be the value of uw. If u is an integer, we can carry out the integration. 


I, § 20] EQUATIONS OF HIGHER ORDER 41 


If uw is a positive integer, the curve has no infinite branches, but it may have 
two forms that are very different in appearance, according to the character of p. 
If » is an odd integer, x is a periodic function (Part I, § 16), and the curve 
I is an algebraic closed convex curve. If uw is even, x increases by a constant 
quantity different from zero when a@ increases by 277; y is always positive. 
We have a periodic curve with an infinite number of cusps on the z-axis. The 
appearance of the curve is that of a cycloid ; it isa cycloid for uw = 2. 


Note. In the examples which we have just studied we always try to reduce 
the integration of a differential equation to the integration of an equation of 
lower order. However singular it may appear at first sight, the reverse process 
may sometimes succeed. Given, for example, an equation of the first order 
S(x,y, y’) = 9, by combining with it a second equation obtained from it by 
differentiation, we obtain an infinite number of equations of the second order 
which are satisfied by all the integrals of the original equation. Suppose that 
we can find thus an equation of the second order which is integrable, and let 
y = ¢(a, C, C’) be the general integral. All the integrals of the original equa- 
tion of the first order are included in this expression, but since they depend 
upon only a single arbitrary constant, there must be a relation between the 
constants C, C’. In order to obtain it, it suffices to write the condition that the 
function ¢(x, C, C’) satisfies the original equation of the first order; we are 
thus led to a certain number of relations between the constants C, C’, and these 
relations should reduce to a single one. 

A most interesting example of this device is due to Monge, who made use of 
it to find the lines of curvature of an ellipsoid. Let 2a, 2b, 2c be the three 
axes ; the projections of the lines of curvature on the plane of the major axis 
and the intermediate axis are determined by the differential equation 


Azyy? + (x2 — Ay? — B)y’ — zy = 0. 
(89) ee a? (b2 — c?) ‘ey ee a?.(a% — b?) 
b2 (a2 — c?) POTS) 
Differentiating the equation (89), and then eliminating the expression 
xz*— Ay? — B, 
we obtain the differential equation of the second order, 


4f 7 if: 
v +5 --=0; 
yoy 


x 
whence we derive first yy’ = Cx, then y? = Ca? + C’. 

The general integral of the equation (89) will be obtained by establishing 
between O and C’ the relation ACO’ + OC’ + BC = 0, as is seen by replacing y? 
by Cx? + C’ on the right-hand side.* 


* The equation (89) can also be easily integrated by the classic processes. It suffices, 
in fact, to put x? = X, y? = Y, after having multiplied all the terms by xy dx?, in order 
to transform it into the Clairaut form. 

Lagrange and Darboux have employed similar devices to integrate Euler’s equation 
(see J. BERTRAND, Traité de Calcul intégral, pp. 569-572). We can also regard a cer- 
tain theorem of Appell’s as an illustration of the same procedure (Comptes rendus, 
Nov. 12, 1888). 


49 ELEMENTARY METHODS OF INTEGRATION | [I, Exs. 


EXERCISES 


1. Find the differential equation of all conics by starting from the general 
unsolved equation and eliminating the coefficients between it and the rela- 
tions obtained by five successive differentiations. 


2. Integrate the differential equations 


Yy?—yP=yy?+y), y(t 2) + zy = 0, 

I+ yyy" =(8y2—-)y4%, @t+y)y’— yy? + wy? + cy’ —y=0, 
ay? + 2ey(y—2ayy—2y(y—2ay=0, = wyy” + ay? — yy’ = 0, 
y?+3y?+y—4=0. 

3. Apply the general methods of depression to the integration of the differ- 
ential equation of conics. 

4, Find the integrals of the equation y” = 2y?(y—1) which are rational 
functions or simply periodic functions of the variable. 

[ Licence, Paris, 1899.] 

5. Given a triangle ABC and a curve IP in its plane, let a, b, c be the points 
of intersection of the sides of the triangle with the tangent at m to the curve I’. 
Find the curves © for which the anharmonic ratio of the four points m, a, b, ¢ 
is constant when the point m moves on one of them. 

The anharmonic ratio of the tangent at m and the straight lines mA, mB, mC 
is also constant. 

6. Given a point O and a straight line D, find a curve such that the portion 
of the tangent MN included between the point of contact M and the point of 
intersection N of the tangent and the line D subtends a constant angle at O. 

[ Licence, Besancon, 1885. ] 

7. Find the projections on the zy-plane of the curves lying on the paraboloid 
2az = mx? + y?, whose tangents make a given constant angle y with the axis Oz. 

[ Licence, Paris, 1879.] 


8. Find the orthogonal trajectories of each of the families of curves repre- 
sented by one of the following equations : 
y? (2a — 2) = 2, y? + mx? — 2ax = 0, 


(a? + y?)? = a®ay, x? + y* = a* log () , 
where a is the variable parameter. 


9. In order that the equation @ (x, y) = C shall represent a family of parallel 
curves, it is necessary and sufficient that we have 


(=) + (F)=9@, 


where ¢(@) is any function of @. 
[Write the condition that the orthogonal trajectories are straight lines. ] 


10. Find the necessary and sufficient condition that the integral curves of 
the equation 7’ = f(z, y) form a family of parallel curves, and show that the 
integration can be carried out by a quadrature. 

[ Licence, Paris, 1898. ] 

11*. Form the general equation of the conics which cut a given conic C 
orthogonally at the four common points. These conics form, in general, several 


1, Exs.] EXERCISES 43 


distinct families. Find the orthogonal trajectories of each of these families, 
Hence derive all the orthogonal systems of which the two families are made up 
of conics. [If f= 0, @¢ = 0 are the equations of two conics cutting each other 
orthogonally at each of their four common points, we have an identity of the 
form 

af Ob , af a9 


SENN, 
Ox on = 8ay Oy PF Mos 


where \ and yp are two constant coefficients. ] 


12. Find the condition that the integral curves of the differential equation 
y = f(x, y) form a family of isothermal curves, and show that an integrating 


factor can be found. 
| [Sopnus Liz. ] 


13. Let y,, y, be two particular integrals of Riccati’s equation (26) (§ 7). 
Show that the substitution (y— y,)/(y — y,) =z reduces the equation to the 
linear equation 

z+ X(y, — Ye)% = 9. 

14, Find a plane curve C such that the triangle formed by any point M of 
the curve, the corresponding center of curvature, and the foot of the ordinate 
of the point M, has a constant area. Show that one of the codrdinates can be 
expressed as a function of the other by a quadrature, and that we can obtain a 
knowledge of the form of the curve without having the definite equation. [The 
axes of codrdinates are supposed to be rectangular. ] 

; [Licence, Paris, 1877.] 

15. Given a plane curve C, let M be any point of that curve, P the center of 
curvature of the curve at the point, and MT the tangent. Through the point 
T where the tangent cuts the axis of x, draw a straight line parallel to the axis 
of y, meeting the normal MP in a point N. Determine the curve C so that the 
ratio of MP to MN is constant. 

[ Licence, Toulouse, 1884. ] 
* 16. Determine the surfaces of revolution such that in each of their points 
the radii of curvature of the principal sections are directed in the same sense 
and have a constant sum a. Sketch a figure of a meridian of the surface. 
[Licence, 'Toulouse, 1878. ] 


17*. Show that the general integral of Euler’s equation can be written in 


the form Ve 
VX —VY 
oes — A(% + y)’— a (@ + Y) — a, = ©, 
t—Y 
where X = a,c* + a,2° + a.a* + a,2 + a, and where Y has an analogous 


meaning. 
° [ LAGRANGE. } 


[It suffices to solve the equation (58) (§ 14) with respect to the constant. 
After a few transformations we obtain Lagrange’s form. ] 


18. The asymptotic lines of the surface represented by the equations 
z= A(u— a)™(v— a)", 
y = Biu— b)m(v— dr, 
Z= C(u— c)™(v— c)m 


44 ELEMENTARY METHODS OF INTEGRATION _ [I, Exs. 


are obtained by the integration of Euler’s equation when we have m=n or 
m+n=1. Deduce from this result the asymptotic lines of the tetrahedral 


surface 
(Gy eer— 
a \b Cc 


19. How can we determine whether a differential equation 
dy — f(x, y)dz = 0 


has an integrating factor of the form XY. , where X depends only upon z, and 
Y depends only upon y, and find this integrating factor when it exists ? 
[ Licence, Paris, October, 1902.] 


20*. Given a plane curve C, the middle point m is taken of the cord MM’ 
which joins any two points M, M’ of that curve. The point M remaining fixed, 
if the point M’ describes the curve C, the point m describes a similar curve c. 
Prove that the curves c satisfy a differential equation of the first order, which is 
integrated, like Clairaut’s equation, by replacing y’ in it by an arbitrary con- 
stant. (Bulletin de la Société mathématique, Vol. XXIII, p. 88.) 


21. Integrate the differential equation 
+7 / y / ec? 
F(y »y — wy”, y— zy +Sy")=0. 


We observe that y’” appears as a factor in the derivative of the left-hand 
side. There exist equations of an analogous form and of any order (see Drxon, 
Philosophical Transactions, Vol. CLXXXVI, Part I; Rarry, Bulletin de la 
Société mathématique, Vol. XXV, p. 71; Bounrtzxy, Bulletin des Sciences 
mathématiques, Vol. XXXI, 2d series, p. 250), 


CHAPTER II 
EXISTENCE THEOREMS 


The first rigorous investigations to establish the existence of the 
integrals of a system of ordinary differential equations or of partial 
differential equations are due to Cauchy. That illustrious mathe- 
matician gave for analytic equations a type of demonstration based 
on a method of comparison to which he gave the name of “calculus 
of limits” (calcul des limites). We owe to him also another method 
which does not assume the functions to be analytic, and which we 
shall discuss later. 


I. CALCULUS OF LIMITS 


21. Introduction. The fundamental idea of the calculus of limits 
consists in the use of dominant functions. The reasoning is quite 
analogous to that which has already been used to establish the 
existence of implicit functions (I, § 193, 2d ed.; § 187, 1st ed.). 
Since every analytic function has an infinite number of dominant 
functions, we see that the method can be varied in a great many 
ways. The simplicity of the demonstrations depends largely on the 
choice of the dominant functions. Since the work of Cauchy, his 
proofs have been perfected and extended to more general cases by — 
Briot and Bouquet, Weierstrass, Darboux, Méray, Riquier, Madame 
Kovalevsky, and many others. Even to-day we make use of this 
same method constantly to treat analogous questions relative to par- 
tial differential equations with various initial conditions. 


22. Existence of the integrals of a system of differential equations. 
Let us consider first a single equation, 


Ot. “U = f(x, 9), 


the right-hand side of which, f(a, y), is an analytic function in the 
neighborhood of a system of values x,, y,. We propose to prove that 
this equation has an integral y(x) analytic in the neighborhood of the 
_ point x, and reducing to y, for x = x,. 

45 


46 EXISTENCE THEOREMS (II, § 22 


Let us suppose for the sake of brevity that x, = y, = 0, which 
amounts simply to writing x and y in place of « — a, and y — y. 
If the given equation has an integral which is analytic in the neigh- 
borhood of the point x = 0, and which vanishes with x, and if we 
can calculate the values of all the successive derivatives of that 
integral for « = 0, we can write the development of that integral in 
a power series. 

The equation (1) gives us first of all (dy/dx), = f(0, 0). On the 
other hand, the equations which we derive from it by repeated dif- 
ferentiations enable us to calculate the value of a derivative of any 
order in terms of x, y and of derivatives of lower order, 


diy ef Of dy 


ees See, 


dx” dy dx 
(2) ay OF 5 Afi Of a her of d*y y 
da® 0x? hy v dz in, Oy? Cy da?’ 


Setting in these relations x = y = 0, we calculate step by step the 
initial values (d*y/dx”),, (d? y/da*),,---, (d"y/da”),, --- of the succes- 
sive derivatives of the desired integral in terms of the coefficients 
of the development of f(a, y) in a power series in a and y. Until 
Cauchy’s work appeared, mathematicians had assumed without proof 
that the power series thus obtained, 


2 (eae eyes ae 
(3) y= (TF + det) 2.1 pees Jaen) | 


was convergent for values of # near zero. 

To establish rigorously this essential point, let us observe that the 
operations by which we calculate the coefficients of the series (3) 
reduce precisely to additions and multiplications alone, so that the 
value obtained for (d"y/dx"), can be written in the form 


d”y 
(4) (72) = Ral Oe Bors B92 °° *y Mond? * *9 Ano) 


where P,, 1s a polynomial with positive integral coefficients, and where 
a, 18 the coefficient of «*y* in the development of f(a, y). If, then, we 
replace the function f(a, y) by a dominant function $(a, Y), and if 
we seek to determine an analytic integral of the auxiliary equation 


(5) — = $(2, Y) 


vanishing with a, the coefficients of the series obtained for the devel- 
opment of Y will be positive numbers greater than the absolute value 


II, § 22] CALCULUS OF LIMITS 47 


of the corresponding coefficients of the same rank in the series (3). 
If the series obtained for Y is convergent in a certain neighborhood, 
the same must be true a fortiori of the series (3). Now the series 
obtained for Y will certainly be convergent if the auxiliary equation 
has an analytic integral vanishing for « = 0. 

Let us suppose that the function f(a, v) is analytic when the varia- 
bles « and y remain in the circles C, C' of radii a and 6 described in 
the planes of the two variables about the two origins as centers, and 
that it is continuous on the circumferences, and let M be the upper 
limit of | f(a, y)| in this neighborhood. We can take for the domi- 
nant function Mu 

oa, Y)= 


Ss) 


and, multiplying the two sides by (1—Y/b), we may write the 
auxiliary equation (5) in the form 
Maye M 


b] 


We can show directly that this equation has an analytic integral 
which vanishes for x = 0. In fact, separating the variables, we obtain 
the integral of that equation in the form 


2 


Y x 
| De eee | er | 
(7) Y 5 uM Log (1 “) 


The constant which must be added to the right-hand side to 
express the general integral of the equation (6) is here zero if we 
adopt for the determination of the logarithm the one which is zero 
for x = 0. Solving equation (7) for Y, we get 

(8) vad —b]1+ 20% Log (1-2), 

If we take for the radical the determination which reduces to 1 
for «= 0, the result (8) represents precisely an integral of the 
equation (6) which is zero for «= 0. This function Y is also ana- 
lytic in the neighborhood of the origin, for the function under 
the radical is analytic in the interior of the circle C of radius a, 
and is zero for 


b 
(9) r=p= alt = a) 
When the variable x remains in the interior of the circle C, of 
radius p described about the origin as center, the absolute value 


48 EXISTENCE THEOREMS [II, § 22 


of (2aM/b) Log (1 — w/a) remains less than unity,* and the radical 
is an analytic function of x in this circle. The series obtained for 
the development of Y is therefore convergent in the circle of radius p, 
and the same is true a fortiori of the series (3) first obtained. 

It is easily seen from the formula (8) that all the coefficients of 
the development of Y are real and positive, a fact which is evident 
also a priori. If we give to x any value whose absolute value is less 
than p, the absolute value of Y will be less than the value obtained 
by replacing # by p. We have, then, for every point in the circle 
C,, |Y|< 6, and therefore |y|< 6. If we replace y in f(a, y) by the 
sum of the series (3), the result of the substitution is therefore an 
analytic function ®(a#) in the circle of radius p. From the manner 
in which we have obtained the coefficients of the series (3), the two 
functions ®(@) and dy/dx are equal, as well as all their successive 
derivatives for «= 0. Hence they are identical, and the Bae 
function y satisfies all the given conditions. 

In order to calculate the coefficients of the series (3), we can substi- 
tute directly for y in the equation (1) a power series y=C a+ C,a7+-- 
and write the conditions that the two sides are identical. The soot 
cient of «”—1in dy/dx is nC,,, while the coefficient of «"~1 on the right 
depends evidently only on C,, C,,-+-, C,_, and the coefficients a,. 
It is easily seen that the coefficients C,, are calculated in this way 
by the use of the operations of addition and multiplication alone. 

The method can be extended without difficulty to a system of any 


number of differential equations of the first order. Let 
‘ 


Fe Ie, Yy» Yo) 855 Yn) (2 =1, 2, -»s) 1) 


be a system of differential equations in which the functions /, are 
analytic in the neighborhood of the values ,, (y¥,),)-++, (Yn), These 
equations have a system of integrals analytic in the neighborhood of 


the point x, and taking on the values (Y,)) (Yo)or** +» Yndo respectively 
for 2 = «,. 

The proof of this theorem can be made to depend on the fact 
that the system of auxiliary equations 


ANG dL Ve Ley eee M 
ea 2 ene Ey See _ 
a b b 


*In fact, all the coefficients of the development of that function in powers of x 
are real and negative. The absolute value of the preceding expression for |x|< p is 
therefore less than its absolute value when x = P, that is, less than unity. 


II, § 22] CALCULUS OF LIMITS 49 


has a system of integrals which are analytic in the neighborhood 
of the origin, and which vanish for « = 0. The functions jf; are sup- 
posed to be analytic as long as we have |x — x,|Sa, |y, —(y,),|=4, 
and M denotes again the maximum absolute value of the functions 
J, in this neighborhood. These integrals, having their derivatives 
equal and all vanishing for x = 0, must be identical, and it suffices 
to consider the single equation 
Dp s M 


# 4-20-27 


in which we can again separate the variables. This equation has 
the integral 


n+1 
b a 


which is analytic in the circle with the radius 


=F 
= alt me (arom) 
and which is zero for « = 0. Hence the system (10) has a system 


of integrals that are analytic in the same circle. 
A single equation of the nth order, 
(12) > eet EE Hie. be geek ; 


ean be replaced by an equivalent system formed of m equations of 
the first order, 


dy dy 
ae OD Mpa 
oy AYn — 2 hYn—1 
om = Yn-1) de = F(a, Y Yt" Nb 


by introducing as auxiliary dependent functions the successive 
derivatives of y up to the (7 —1)th order. We deduce from the 
general theorem, then, the proposition that the equation (12) has an 
analytic integral in the neighborhood of the point x, and such that 
that function and its first n —1 derivatives take on for x= a, the 
values Y,, Yoy+ ++, YY gwen in advance, provided that the function 
F is analytic in the neighborhood of the system of values &,, Yo) Yoo 
66, Y@rY, | 

From the demonstration it results that there cannot be more than 

one analytic integral of the equation (1) taking on for « =a, the 


50 EXISTENCE THEOREMS [II, § 22 


value y,. But nothing enables us to say up to this point that there 
do not exist non-analytic functions satisfying the same conditions.* 
This is a point which will be rigorously established farther on 


(§ 26). 


23. Systems of linear equations. We shall find farther on, by another 
method, a larger value for a lower bound of the radius of convergence 
of the series which represents the integrals ($ 29). If the functions 
Ff; have special forms, we can sometimes employ more advantageous 
dominant functions, still making use of the method of the calculus 
of limits. 

In particular, this is what happens in the very important case of 
linear equations. Let 


(14) Hany tate t+ ente +  G=1,2,-+42) 
be a system of linear equations in which the functions a;, and 0; are 
functions of the single variable x, analytic in the circle C of radius R 
about the point x, as center. These equations have a system of inte- 
grals analytic in the circle C and reducing respectively to (Y,)5) (Yo)o 
-» Yn)y for & = 2). 
We may suppose in the proof that 


(Yo aie. (Y)o ae Ole oie (Yn)o = 0, 


for if we change y; into (y;), + y;, the system (14) does not change 
in form, and the new coefficients are again analytic in the circle C. 
Let M be the maximum value of the absolute values of all the 


* The following is the reasoning used by Briot and Bouquet to treat this matter. 
Let y; be an analytic integral of the equation (1) taking on the value ¥9 for x= Xp. 
Putting y=y,+ 2, the equation (1) takes the form 


(v) = ZW (2, 2), 


where Y (x, z) is analytic for x=2% 9, z=0. Let us suppose that this equation has an 
integral, other than z=0, approaching zero when the variable x describes a curve C 
ending in the point %). Let x1, v_ be two points of this curve to which correspond the 
two values z, and 2, of z. We obtain from the equation (1’) 


foes fs V (x, z) dx. 


If x; approaches x9, 2; approaches zero, and the absolute value of the left-hand side of 
this equality becomes infinite, while the absolute value of the right-hand side remains 
finite; there cannot be, then, an integral approaching zero different from z=0. But 
the reasoning supposes that the point x approaches xy along a curve C of finite length. 


II, § 24] .CALCULUS OF LIMITS ab 


functions a@;,, 6; in a circle C' with the center x, and the radius 7 < R. 


The function 
M 
———— (14%, + FF + Y,) 
4—2%=% 


‘he 


is a dominant function for all the functions a,,y, + +--+ any, + di, 
and we are led to consider the auxiliary system 


HAE ye S| ea at BAe j 
a) Ta et ae, ig ea Gs cet 
r 


Since the functions Y,, Y,,---, Y, are required to be zero for 
x = a,, and since their derivatives are equal, they are identical, and 
the system (15) can be replaced by the single equation 

oy, M 
(16) Ne napeta hate) 
0 


which can be integrated by separating the variables. The integral 
which is zero for « = a, has the form 


a —nMr 
ae (1 Hs wo) de 1] 
nv fe 


and it is analytic in the circle C’. The same thing is therefore true 
of the integrals of the system (14), and, since the number 7 may 
be taken as near & as we wish, it follows that these integrals are 
analytic in the circle C. | 


' 24, Total differential equations. Let 7,, x,,---, Z, be a system of n independ- 
ent variables, let z be an unknown function of these variables, and let f,, /, 

- +, f, be n given functions of 2,, %,-+--, Zn, z. A total differential equation is 
a relation of the form 


(17) dz = f,dz, + fdr, + --- + fran ; 
it is really equivalent to n distinct equations: 


OZ. 


; Oz Oz 
18 —=—)J;, —$S = eae, ——— re 
oy fi Ot, ts On I 


Ox, 
Let us suppose that there exists a function z of 7,, a, --+, &, satisfying 
these n relations. We can calculate the second derivative 02z/dxr;dxr, (i #4 k) in 
two different ways. Writing the results obtained as identical, we obtain thus 
n(n — 1)/2! relations of the form 
Ofk , Of >» 


OF} R104: . 
Bip ces SRM a (i, k= 1, 2,+++, n) 


(19) 
OLE, 0% On; = 


o2 EXISTENCE THEOREMS [II, § 24 


and the function z can only be taken from among those functions which satisfy 
these relations. We are going to consider only the very important case, in 
which these relations are satisfied identically. The equation (17) or the equiva- 
lent system (18) is then said to be completely integrable. 


Given a completely integrable total differential equation in which the functions fi 
are analytic in the neighborhood of the system of values (X1)o, (Le)os***s (Ln)o» Zo» 
this equation has an analytic integral in the neighborhood of the system of values 
(©1)o.***s (Cn)o, which reduces to Z when ©, = (©1)oy ++ +s Tn = (Ln)o- 


. The equations (18) and those which are derived from them by successive 
differentiations enable us to express all the partial derivatives of the unknown 
function z in terms of 2, 7, £,-++, Z,»; hence we can obtain the values of the 
coefficients of the development of the analytic integral, if it exists. But, while 
it is evident that we can calculate such derivatives as 0?z/dx? in only one way, 
it requires a little more care to assure ourselves that we shall always obtain the 
same expression for a derivative of any order, such as 6? + 9z/éx? dxf, which can 
be calculated in several different. ways. This will be the case for the deriva- 
tives of the second order, if the conditions (19) are identically satisfied. In 
order to show that the same property is true in general, it suffices to show that, 
if it is true up to the partial derivatives of order p, it will also be true for the 
partial derivatives of order p +1. We shall base the proof on the following 
fact: Let U(a,, £4, +++, %n, 2) be any function of x,, %2, +++, Lp, Z, and let us put 


dia U peas d?U da aUs 


dz; bay oz dx; Axx. 7. dit, 
From the conditions (19) it follows immediately that we have for any function U 
the relation 


(i,k =1, 2, +++, n) 


au du 
da;dxx dx,dxi 


Let now u and v be two partial derivatives of the pth order differing only in 
the fact that a differentiation with respect to z; in one has been replaced by a 
differentiation with respect to zz; in the other. The proof depends on showing 


that we have 
ou Ou ov 


Ov 
or that du/dz;, = dv/dz;. But u and v have been obtained by taking the partial 
derivatives of a partial derivative w of order p — 1 with respect to the variables 
x; and az, respectively. We have therefore u=dw/dz,, v = dw/dzz, and the 
equality to be established reduces to d*w/dz;dx;, = d*w/dx,dz;, an equality which 
has already been proved. 

To prove the convergence of the series thus obtained, we can therefore replace 
the functions f; by dominant functions ¢;, provided that we choose these func- 
tions ¢; so that the resulting auxiliary total differential equation shall itself be com- 
pletely integrable. For simplicity let us put (x,)) = (%_)y = +++ = (€n)p =%p = 03 
we can take for the dominant function of all the functions f/; an expression of 


the form 
M 


Il, § 25] CALCULUS OF LIMITS 53 


and the auxiliary equation 


M (da; + dxg + +++ + dap) 


(.-atet te) (_ 2) 
r p 


is completely integrable from the symmetry of the right-hand side relative to 
the n variables z;. In order to obtain an analytic integral that vanishes with 
these variables, we need only seek an integral which is a function of the single 
variable X = 2, + %,+--+-+a2,. This leads to an ordinary differential equation 


of the form (6) 
(1- |) az ag 


(20) dZ = 


iad 

r 
Since the integral of this equation is represented by a development in a con- 
vergent series the coefficient of any term af1--- pee of which is real and positive, 
the development obtained for z is a fortiori convergent in the same neighborhood. 
The theorem can be extended without difficulty to systems of total differential 
equations in n independent variables 7,, 2, +++, %, and m dependent variables 

219 %q3 °° %s Sm? 
(21) dan =f indy + +++ + Sunday + +++ + Sanditn. (ane sie) 
Se OP re. 


By calculating in two different ways the derivatives of the form 672;,/0x; 02% 
we are led to the conditions 
Ofin , OFin 


(22) sat ie 


Of in 


Om 


_ Often, OFtn » Ok 


Ser tices t+ Fem Sim: 
The system (21) is said to be completely integrable if these conditions (22) are 
satisfied identically, and we have the following theorem which is demonstrated 


like the preceding : 


Every completely integrable system in which the functions f; are analytic in the 


neighborhood of a system of values (£1), (®e)or°°*s (Ln)oy (Z1)o ++ *s (%m)o has a 
system of integrals analytic in the neighborhood of the point (X,)o, +++, (Ln), and tak- 
ing on respectively the values (2)o, (Ze)o9 + +s (Zm)o When L = (L1)o, +++, Fn = (n)o- 


25. Application of the method of the calculus of limits to partial differ- 
ential equations. The calculus of limits enables us also to prove the 
existence of integrals of a system of partial differential equations. 
Let us consider first an equation of the first order, 


(23) ma F(t Hig ¥ 7 Lay ea) 
in which the right-hand side does not contain the derivative ¢z/ém,. 
This equation and those obtained from it by successive differentia- 
tion enable us to express all the partial derivatives of z in terms of 
,,%,,+++,a,,%, and of the partial derivatives of z taken with respect 
to the variables x,, 7,,---, 2, alone. This property is evident for the 


54 EXISTENCE THEOREMS [II, § 25 


derivatives of the form 0%+:''++1z/0x, Cxgz.-- dxf", as is seen by 
differentiating the two sides of the equation (23) a, times with 
respect to #,,---, and then a, times with respect to ,. If we differen- 
tiate the two sides of the equation (23) once with respect to x,, and 
any number of times with respect to the other variables x,, x,, +++, Xn; 
and if we then replace in the right-hand side of the result the par- 
tial derivatives which involve just one differentiation with respect 
to the variable x, by the expressions already obtained, we shall obtain 
also the derivatives O%t+ "+ ¢"+2z /0x?0xg2..-+ Cxa" expressed in the 
manner stated above, and it is clear that we can continue to apply 
the same process indefinitely. 

Let us now suppose that the function / is analytic in the neigh- 
borhood of a system of values (@,),, +++, (®n)o» 22 (Dodoo °° *» (Dados 
and let $(x,, #,,---, ,) be a function of the (mn —1) variables 
I, Hz, +++, #, analytic in the neighborhood of the point* (@,),, 
(a5)o9 ***y (,), and such that we have for these particular values 


= (F)=Gdn (FE)=@dn o> (FE\=@oe 


If these conditions are satisfied, the equation (23) has an integral 
which is regular in the neighborhood of the point (3,),) +++) (@n)y and ~ 
which reduces to p(X.) %, +++, L) for L, =(2X,)p. 

By hypothesis, the function $(#,, x,,-+--, x,) can be developed in 
a series of positive powers of the variables x; —(#;),, and the coeffi- 
cients are, except for certain numerical factors, the values of the 
partial derivatives of that function at the point (a,),, (a5), +++) (@n)o: 
Since the function z, the existence of which we wish to prove, must 
reduce to $(%,, 7, +--+, x,) for x, =(#,),, we know from that fact 
alone the values at the point (@,),, (@,),, +++; (@,)) of all the partial 
derivatives of the function z which involve no differentiation with 
respect to the variable x,. We have just seen how all the other partial 
derivatives of z can be expressed in terms of these. We can there- 
fore calculate, step by step, all the coefficients of the development of z 
according to powers of x; —(a,), in terms of the coefficients of the 
two developments of the function f and of the function ¢, and the 
calculation involves the operations of addition and multiplication 
alone. We can therefore employ again dominant functions to prove 
convergence: if the series obtained by replacing, in the preceding 


* For the sake of brevity we shall designate as a point every system of particular 
values, real or imaginary, assigned to the variables appearing in the discussion. 


Ii, § 25] CALCULUS OF LIMITS 55 


calculation, f by a dominant function /’, and ¢ by another dominant 
function ®, is convergent, the same thing must necessarily be true of 
the series obtained for z. 

We can, first of all, replace the given initial conditions by other 
simpler conditions by means of a succession of easy transformations. 
We may suppose (a), =(a,), = ++: =(#,), = 9, for that amounts to 
writing «,; in place of x; —(a;),. If we also put 


% = P(Xq) Ly, ++) Ly) + u, 


the new unknown function « must reduce to zero for x,=0. We 
may suppose also that after these transformations the right-hand 
side of the equation, when developed, does not contain a constant 
term, for if the development commenced with a constant term a 
different from zero, it would suffice to put « = ax, + v in order to 
make it disappear. Having made these transformations, if we now 
replace the right-hand side by a suitable dominant function, the 
demonstration of the theorem reduces to showing that the equation 

OZ M 

(,atatotata(, & 2) 
r p 

where M,7, p are determined positive numbers, has an integral which 
is analytic in the neighborhood of the origin and which reduces to 
zero for z,=0. If we replace x, on the right-hand side by zx,/a, 
where @ is a positive number less than unity, we increase the coefii- 
cients, and the theorem will be established a fortiori if we prove 
the proposition for the new equation 


OZ M 
7 OZ OZ 
=i Se ee eet geet LRU RLY ae 
yeti eam Sal Rael Ur pee ta El ed 
r p 


Indeed, it is sufficient to show that this equation has a regular 
integral, represented by a power series whose coefficients are all real 
and positive; for the coefficients of this third development are at 
least equal to those of the series obtained by supposing that 7 van- 
ishes when 2, = 0, since the coefficients are all obtained by means 
of additions and multiplications of the coefficients of the terms inde- 
pendent of x, In order to establish this last point, let us try to 
satisfy the equation (25) by taking for Z a function of the single 


56 EXISTENCE THEOREMS [II, § 25 


variable X = 2,/a+a,+---+,. We are thus led to the differen- 
tial equation of the first order, 


5 eb ie Bers | C2 “Nala M 
(26) Ca ee ne eee wae 
Ife 


Let us suppose that @ has been chosen so small that the coefficient 
of dZ/dX on the left is positive. For X = Z = 0 the equation (26) 
has two distinct roots, one of which is equal to zero. That equation 
has therefore an analytic integral in the neighborhood of the origin, 
which, together with its first derivative, is zero for X = 0. It is easy 
to show directly that all the coefficients of the development of this 
integral are positive; for the equation (26) may be written in the 


form 
aZ gs a(S 


2 
dx ) t+ ®(X, 4); 


where A is positive and where ®(X, Z) denotes a series whose coeffi- 
cients are all positive. After a first differentiation we find 
PZ daZa@Z  0&  0®dZ 


qe 74 ox aetixtig a 


For X = 0, Z and dZ/dX are zero; hence d?Z/dX? is positive. The 
verification for the following derivatives is similar. 

The series obtained for the development of the desired integral z 
is therefore convergent as long as the absolute values of the differ- 
ences x; —(x;), remain less than a positive number 7. The value of 
that series is an analytic function in the neighborhood of the point 
(X,)o. (Lalor °**» (ln) and reduces to'd(x,, &,,---, x,) for x, =), 
That function satisfies the given equation; for if we replace in f the 
variables z, 0z/0x,, ---, 0z/0x, by the preceding function and by its 
partial derivatives, the result is a function (a, x,,---, a,) which 
is regular in the neighborhood of the point (@,),, (@,).) +++, (nos 
and, from the manner in which we have obtained the coefficients 
of the series z, the two functions y and @z/éx, are equal, as well as 
all their partial derivatives, at the point (#,),, (@,)))---+) (@n)9 They 
are therefore identical. 

The proof is the same for a simultaneous system of equations of 
the first order, 

02, 


(27) ae, TY on, IY aieiag math. 


1 


II, § 26] CALCULUS OF LIMITS 57 


whose right-hand sides contain only the variables a,, x,,---, x,, the 
functions 2,, 2,,-+-+, #), and the partial derivatives of the first order 
except those with respect to x, Supposing the right-hand sides ana- 
lytic in the neighborhood of a system of particular values («,),, (2,),, 
(p*),, assigned to all the variables which appear in the function f, 
these equations have a system of integrals which are analytic in the 
neighborhood of the point (@,),, +++, (&,), and which reduce for 
x, =(2,), to p gwen functions $,, ,, +++, by of the (n —1) variables 
Loy yy ++ *y Lp, Which are analytic in the neighborhood of the point 
(2)or (®y)or °° *y (Hn), and are such that the values of $, and of 0p,/0x; 
at that point are precisely (z,), and (pf), (k=1, 2,---, p; i= 2, 
Se hin), 


26. The general integral of a system of differential equations. The 
preceding theorem enables us to complete the theory of differential 
equations on several important points. Thus, the existence of an 
infinite number of integrating factors for an expression of the form 
P(a, y)dx + Q(x, y)dy is an immediate consequence of it if P and 
Q are analytic functions of the variables « and y (§ 12). 

Let us consider again the equation of the first order y' = f(a, y), 
and let(«,, y,) be a pair of values for which the function f(a, y) 
is regular. The analytic integral the existence of which has been 
established, which takes on the value y, for =a,, may be con- 
sidered as a function of three independent variables x, x,, y,; it is 
from this point of view that we are going to study it. For definite- 
ness let us suppose that the function f(x, y) is regular in the 
neighborhood of a point (« = a, y= 8). We can evidently consider 
the given equation as a partial differential equation, 


(28) cL = #(a, 9), 


which defines a function y of the three variables x, x,, y,, and we 
propose to determine an integral of that equation which is analytic 
in the neighborhood of the point =a, x,=a, y, = 8 and which 
reduces to y, for «=«a,. This last condition is not in the same 
form as that of the preceding paragraph, but it suffices, in order 
to overcome the difficulty, to take instead of aw and of x, two 
new independent variables u=a+a, and v=a—zx, Then the 
equation (28) becomes 
(29) ou 4 U(X", y), 


Ov 


58 EXISTENCE THEOREMS (It, § 26 


and we are led to seek an integral of this new equation which is 
analytic in the neighborhood of the values u= 2a, v=0, y¥,=B 
and which reduces to y, for v=0. By the general theorem, there 
exists an analytic integral, and only one, which satisfies these 
conditions ; we shall denote it by (a, x,, y,), supposing that we 
have replaced w and v by their values in terms of w and y. Let D 
be a region defined by the conditions |«—a|=r, |x,—a|=7, 
ly. — 8| =p, in which the function (2, x,, y,) is regular. The 
function @ has the following properties in this region. In the first 
place, from the very way in which we have obtained it, if x, and y, 
are constants, it represents the integral of the differential equation 
y' = f(x, y), which takes on the value y, for «=ax,. This integral 
is surely analytic whenever | — a@| is less than r, for any point 
(x), Y,) 1 the region D. 
The development of ¢(a, x,, y,) is of the form 


Y = Yo t+ (@— X) PM, Hy, Yo), 

where P also denotes a regular function. By the general theory 
of implicit functions, we can solve the above relation, obtaining 
Y, = (a, #, y), in which the right-hand side is also a power series. 
The function (a, x,, y) ts identical with o(x,, x, y). In fact, let a, 
and x, be two values of x in the region D; then the integral which 
is equal to y, for « = a, takes on at the point w, a certain value y,, 
and we have y, = $(x,, %,, y,). But it is evident that the relation 
between the two pairs of values (x, y,), (@, y,) 18 a reciprocal one; 
hence we have also y, = $(&,, @y Y,): 

Let x; be any value of x such that we have |aj—a|<r. Every 
analytic integral of the equation (28), passing through any point 
(2 Y,) of the region D, satishes a relation of the form 


(30) Play e, Y) —— Ce 

For, let us consider the analytic integral equal to y, for «= ~a,. 
That integral takes on a value yj) when a has the value x), and we 
have, from the definition of the function ¢, ¢(x;, x, y,)= yj. Let 
be another value of the variable in the same region and y the corre- 
sponding value of the integral. We have also $(a, a, y) = y;, and 
therefore the analytic integral considered does satisfy a relation of 
the form (30). By differentiating it with respect to x and replac-. 
ing y' by its value f(@ y) we find that the function (a, a, y) 
satisfies the relation 


0 0 
(31) bet gy fv) = 0. 


II, § 26] CALCULUS OF LIMITS 59 


This relation reduces necessarily to an identity, for it must be true 
for x=2,, y= ¥Y, and the point (x,, y,) is any point of the region D. 

This enables us to answer a question left undecided in § 22. 
In the plane of the variable x let any curve I approach the point a, 
as a limit. We shall say that a function y of the variable « which 
can be continued analytically along the whole length of I approaches 
y, as x approaches x, on Tif for every positive number e we can find 
a corresponding positive number y such that |y— y,| remains less 
than e for all the values of x lying on rin the interior of a circle 
with a radius 7 and with the center z,. 

The reasoning of Briot and Bouquet does not prove that there do 
not exist other integrals than the analytic integral, approaching y, 
as « approaches x, in the manner which has just been defined. This, 
however, is the fact. For let us consider a definite point (x,, y,) 
of the region D, and let us take for the new dependent variable in 
the equation (28) the function Y= ¢(a,, «, y) defined above. Then 


we have dY a4 ag dy 
du Oy dx’ 


and, by the relation (31), the given differential equation reduces to 
dY/dx =. If, now, y approaches y, when x approaches @,, the same 
thing is true of Y, and the only integral of the new equation dY/dx = 0 
which satisfies this condition is evidently Y= y,. The integral sought 
must therefore satisfy the relation 


(2X, wv, y) = Yor 
or 


(32) Yy=Y t+ (eX) P(x, Y, X), 


and, by the theorem on implicit functions (I, §193, 2d ed.; § 187, 
1st ed.), there is only one root of the equation (32) approaching y, 
as x approaches @,, and that root is an analytic function.* 

It follows that every integral of the equation (28) which passes 
through a point of the region D satisfies a relation of the form (80). 
On that account we say that that equation represents the general 
integral of the differential equation in this region. The number C 
is the constant of integration which remains arbitrary at least be- 
tween certain lmits. We have seen that we could also put the 
equation (30) in the equivalent form y= (a, 2, yj), where the 
constant of integration is ¥. 


* PicarD, Traité d’ Analyse, Vol. I, pp. 315-317. Paintevh, Legons de Stockholm, 
p. 394. 


60 EXISTENCE THEOREMS [II, § 26 


All these properties can be extended to a system of differential 
equations of the form 


“th dy dy, 
(33) =~ : =f,(@; 9p Far 4 Yn) ais = Sy a wt =f. 


Let us suppose that the right-hand sides are analytic in the neigh- 
borhood of the system x =a, y, = 8, --++, Y= 8B, We may again 
regard the preceding equations as a system of partial differential 
equations involving the m dependent variables y,, y,,---, y, and 
the n + 2 independent variables x, x,, (Y,)o. (Ya)o. °° *» (Yn)or and we 
may seek the integrals of this system which are regular in the 
neighborhood of the values x = a, x, =a, (Y,), = By -*+» (Yn)p = B 
and which reduce to (¥,),; (Yz)o. +++» (Yn)o Tespectively for « = a. 

Let 
(34) pea $,[2, Loy CAN sae (Yn)o]3 Y, — $,; Seay 

Yn = PrL%, Hy» (Y)os ia CAN 


be the n functions thus defined, which we suppose to be analytic 
in the region D defined by the conditions |jx—a|=r, |a,—a|=7, 
\(y:), —B:|==p. From the equations (34) we derive, conversely, 


(35) (Y%)o = $,(%,, LY °* % Wars OR (Yn)o = Pn(Lo UY" %y Yn) 


and each of these functions ¢, satisfies, for any value of «,, the 
relation 


2 
dy, it . aes). 


We prove this just as before by observing that the analytic 
integrals which take the values (y,),,---, (Y,), for «=x, satisfy 
the relations (35), and therefore the relations (36), which we deduce 
from them by differentiating with respect to the independent 
variable « and by replacing the derivative dy,/dx by jf; These 
relations (36) must reduce to identities; for if x, is supposed fixed, 
we can show as above that we can choose (y,),, +++, (Yn), in such a 
way that the integral curve * passes through any given point of the 
region D. The left-hand side of the equation (36) must therefore 
be zero for the codrdinates of any point whatever of this region. 

If in the equations (33) we take for new dependent variables 
the n functions Y;=¢,(2,, &, Y)-++ +; Y,), Where x, is constant, these 


* As a generalization we shall say that every system of integrals of the equations 
(33) defines an integral curve. 


II, § 27] SUCCESSIVE APPROXIMATIONS 61 


equations become, by the conditions (36), 


par rora et say, aY, 
e0) dx a dx ae me dx Gia 


It follows that all the integrals of the system (33) satisfy relations 
of the form (85), where (y,),,---, (y¥,)) are constants — at least all 
of those integrals which have a point in the interior of the region 
D where the functions ¢@ are regular. We shall say, then, that the 
equations (35) represent the general integral of the system (33) in 
this region. 

From these equations it follows also that there are no other 
systems of integrals than the analytic integrals which approach 
(Yo °° *» (Yn)) When x approaches x,. We have, in fact, 


$= Y;+ (x = %,) P;(a%3 BY °**s UP. 


and the Jacobian D(¢,, $,,--+; Gn)/D(Y» Yor +++» Yn) Teduces to 
unity for x =«a,. According to the general theory of implicit func- 
tions, the equations (35) have only a single system of solutions for 
Yy Yor***y Yn» Which approach (y,),,-++, (Yn), When « approaches ~«,, 
and these solutions are analytic. 

To sum up, through every point of the region D there passes an 
integral curve, and only one, represented by n equations y; = y;(a), 
where the functions y; are analytic so long as |x — a|Sr. 


~ 


II. THE METHOD OF SUCCESSIVE APPROXIMATIONS. THE 
CAUCHY-LIPSCHITZ METHOD 


27. Successive approximations. The method of successive approximations has 
been applied with success by E. Picard to ordinary differential equations and 
to a great number of cases of partial differential equations. We shall apply it 
to the treatment of differential equations with an important addition due to 
Ernst Lindel6of. 

Let y(z) be an integral of the differential equation dy/dz = f(z, y) taking 
on the value y, for x= 2z,. The function y (x) satisfies the relation 


(38) y()=+ f SIL vOldt 


and conversely. The equation (88) is an integral equation which is equivalent 
to the two conditions y’ (x) = f[x, y (x)], y (%)) = Yp and which lends itself readily 
to the method of successive approximations. We shall develop the method on 
a system of two equations of the first order 


(39) — Berens), Z=oena, 


62 EXISTENCE THEOREMS [II, § 27 


supposing first that the variables are real. We shall assume that the two func- 
tions f and ¢ are continuous when z varies from «, to z)+ a and when y and z 
vary respectively between the limits (y)— 0, yy + 0) and (%)—c¢, Z + ¢); that 
the absolute value of each of these functions f and ¢ remains less than a posi- 
tive number M when the variables «, y, zg remain within the preceding limits ; 
and, finally, that there exist two positive numbers A and B such that we have 


P(e, ¥, 2)—- F(a, 9’, 2%) |< Aly—y'|+ Ble—@|, 


40 
(40) l(a, ¥, 2) — O(a, y', 2) |< Aly—y|+ Blz—2 | 


for any positions of the points (a, y, z) and (a, y’, 2’) in the preceding region. 

Let us suppose, for ease in the reasoning, a>0, and let h be the smallest of 
the three positive numbers a, b/M, c/M. We shall prove that the equations (89) 
have a system of integrals which are continuous in the interval (%), % + h) and 
which take on the values y, and Z) for £=2,). For this purpose we shall write the 
equations (39) in the form of integral equations : 


(1) v@)=ntf fhyO,2O)d,  2e)=4 fol yO, 2) 4, 


and we shall solve these equations by successive approximations in the same way 
as for a system of simultaneous equations (I, § 34, 2d ed.; § 25 ftn., Ist ed.), 
taking for the first approximation values the initial values y, and z, themselves. 
We are thus led to write 


W(t) =U + f F(t, Yos 0) A, 
aa) 


%1(©) = % + fe (f, Yos %q) at, 
(42) we 
volt) =Yot f FU malt), aO1at, 


z(t) =z tf lt, nO, a())a 
Bg 
and, in general, 


Un (t) =U +f SUts Um —a(l)s en—1(t)] a, 
(43) oy 
Zn (@) = 2 +f [t, Yn—1(t), Zn —1(t)] at. 


Let us prove first that this process of approximation can be continued indefi- 
nitely if x is contained in the interval (x), % +h). We have, in the first place, 
if x is within that interval, 

[Ysa — Yol< Mh <b, 


and, similarly, |z, — z)|<c. If we replace y and z by y, and z, in the functions 
J and ¢, the functions of z thus obtained are therefore continuous between 2, 
and z,+ A, and their absolute values remain less than M. For the same reason 
as before, y, and z, are continuous functions of gz in the interval (x, x + h), 
and we have in this interval |y, — y¥)|<b, |z,— 2% |<e. The reasoning can be 
continued indefinitely ; all the functions y, and z, are continuous between £, and 
ty +h, and we always have in this interval | y, — y9|<b, |Z, — %|<¢, 


Il, § 27] SUCCESSIVE APPROXIMATIONS 63 


In order to prove that y, and z, approach limits when n becomes infinite, let 
us notice that we derive first, from the first of the relations (42), 
(44) lWr(Z)— Yol< M(e@— a), = 24 (%) —%|< M(@@— 2), 


where « is any value whatever except x, in the interval (zr), 2) +h). We have next 
MH 
volt) — wy(2) =f {414 40, AOI — Fb Yo, %0)} ay 
*9 


and, by taking account of the first of the inequalities (40), 
late) — me) |< fo Aly) —volat + f° Blay()— zoldt 
0 0 


and therefore, by the inequalities (44), 


L— Lo)? 

Ivele) — wy(a)|<(A + By MERE, 

We have an analogous result for |z, (x) — z,(x)|, and, continuing in this way, 
we see that we have in general 


| Yn() — Yn—1(@) |< M(A + Baits Sh 
(45) A 
\Zn(2) — Zn—1(t) |< M(A + B)r-1 ce 


The two series 


(46) Yo + (Yy — Yo) +H (Ye — V1) Fees +H Yn— Yn-1) Fees 
Zt (2 — 2) + (ey — 2%) +++ + en — 2n-1) + ey 


whose terms are all continuous functions of x in the interval (x,, 2) + h), are 
therefore uniformly convergent in that interval. The values of these two series, 
Y (x) and Z (x), are consequently continuous functions of « between x, and x + h. 
As the number n becomes infinite, the relations (48) become, at the limit, 


Tey 17a +f fle, Y(t), Z()] dt, Z(t)=zZ, +f¢ [t, Y(t), Z(d)] dt. 


For we have just seen that the differences Y (x) — yn-1(x), Z() — 2n-1(2) 
approach zero uniformly in the interval (x), x) + A), and therefore, by virtue of 
the relations (40), the integrals 


fst YO); ZO] — Slt yo-a(t); 24-10) |} at, 
f ott Y(t), Z(t)]— $[t, Yn—r(t), Zn —1(t)]} dt 


approach zero when n becomes infinite. The functions Y (x) and Z (z) therefore 
satisfy all the given conditions. 

The preceding method is evidently applicable, whatever may be the number 
of the equations in the system. The inequalities (40), which play an essential 
part in the demonstration, are certainly satisfied for suitable values of A and B 
whenever the functions f and ¢ have continuous partial derivatives with respect 
to y and z within the limits indicated for the variables; this is an easy conse- 
quence of the law of the mean (I, § 20, 2d ed.; § 11, Isted.). Let us also notice 


64 EXISTENCE THEOREMS [ II, § 27 


that if the functions f and ¢ remain continuous when @ varies between x, — a 
and x, + a, and the variables y and z between the same limits as above, the same 
reasoning proves the existence of a system of integrals, Y (x) and Z (x), which 
take on the values y, and z, for « =, and are continuous in the interval (x — h, 
Zo + h), where h has the same meaning as before. 

There are no other systems of integrals than Y (x) and Z (x) taking on the values 
Y, and z, for =a). The reasoning being always the same, let us take for sim- 
plicity a single equation dy/dx = f(x, y), and let us put, as before, 


Vy =Yot f Sh yydh Ieee Un =Yot f S[t, Yn—1(6)] at. 
Be et 


Let Y,(«) be an integral of that equation which takes on the value y, for z= 5 
and which is continuous in the interval (x), 2) + a’), where a’ is less than the 
smaller of the numbers a and b/M and such that we have | Y,(x) — yp| <6 in this 
interval. Since Y, satisfies the given equation, we can write 


Yi (©) — Yo =f Flt, Y,O] a, 
X 
and, consequently, 


¥4(c) — yn(e) =f (714 YO] — SLs ma) a. 


Let us put successively in that relation n = 1, 2, 3,---; we have first 


| ¥y (x) — y,(@) |< Ab(x — Zp), 
then 


| ¥4(x) — y9(z) | <A f Ab( SRY, ey Likes = “el, 


and, in general, 
[Yy(0) — vn(a)|<.Am S— Ao", 


The right-hand side of that inequality approaches zero when n becomes 
infinite ; the integral Y, is therefore identical with the limit of y,, that is, 
with Y *. 


28. The case of linear equations. The.general reasoning proves that the integrals 
are certainly continuous in the interval (x), % + h) defined above; but in quite 
a number of cases we can state the existence of a more extended interval 
in which the integrals are continuous. If, in fact, we go over the proof again, 
we see that the conditions h<b/M, h<c/M are needed only to make sure that 
the intermediate functions y,, 21, Yo, Z,°°+ do not get out of the intervals 
(Yo — 0, Yo + 5), qo — ©; %) + ¢), SO that the functions f(x, yi, Zi), O(X, Yi, 2) 
shall be continuous functions of x between x and 7, +h. If the functions 
F(x, y, 2), @(&, y, Z) remain continuous when z varies from @, to x) + a, and 
when y and z vary from — o to + o, it is unnecessary to make these require- 
ments. All the functions y; and z; are continuous in the interval (x9, 2) + a). 


* Regarding questions concerning the approximate integration of differential 
equations, the reader is referred to the articles of E. Cotton (Acta mathematica, 
Vol. XXXI; Bulletin de la Société mathématique de France, Vols. XXXVI, XXXVII, 
and XXXVIII; Annales de I’ Université de Grenoble, Vol. XXI). 


II, § 28] SUCCESSIVE APPROXIMATIONS 65 


Again, in order to prove the convergence of the two series (46) it is sufficient 
that there exist two positive numbers A and B such that the two inequalities (40) 
are satisfied for any values of y, y’, z, 2’ if x remains in the interval (z,, x) + a). 
We recognize, in fact, on going over the calculations made above, that the in- 
equalities (45) still hold, provided that we indicate by M an upper bound of 
IF (X, Yo, %o)| and of | (x, Yo, Zo)| in the interval (x, Lp) + a). 

These conditions are satisfied, according to the law of the mean, if the 
functions f(z, y, 2), (x, y, z) have partial derivatives with respect to the 
variables y and z which remain finite for all values of y and z when z varies 
from £) to + a. Such, for example, is the case for the equation 

dy =x+siny; 

dz 
the right-hand side is a continuous function, whatever 2 and y may be, and 
the partial derivative @f/doy is at most equal to unity in absolute value. 
All the integrals of that equation are therefore continuous functions when « 
varies from — o to + o.* 

The preceding conclusions apply in particular to systems of linear equations 

(47) we = iy; + GieY2t-+++Gnint bi, (i= 1, 2,---, n) 
where the coefficients a, b; are functions of x. If all these functions are 
continuous in an interval (x, z,), all the integrals of this system are likewise 
continuous in this interval; if the coefficients are polynomials, all the integrals 
are then continuous when z varies from — « to + 0. 

Limiting ourselves to real variables, we see that the integrals of linear equa- 
tions can have no other singular points than those of the coefficients. This very 
important property cannot be extended to many other equations, even though 
they are apparently just as simple — for example, to the equation 7’ = y?. 


Note. We often have occasion to study systems of linear equations whose 
coefficients are analytic functions of certain parameters. Let us suppose, for 
definiteness, that the coefficients a, and 6; of the equations (47) are continuous 
functions of # in an interval (a, b), and that they depend also upon a parameter X 
of which they are analytic functions in a region D. 

The integrals of this system which take on given initial values for a value z, 
of « included between a and 6 are represented in the whole interval (a, b) by 
uniformly convergent series, and from the very manner in which we obtain 
them it is clear that all the terms of this series are analytic functions of the 


* We can deduce an analogous theorem from the calculus of limits. Let f(x, y) be 
a function which is real for every system of real values of % and y and analytic in 
their neighborhood. Suppose, besides, that | f(x, y)| remains less than a fixed num- 
ber M when we have respectively |R(x/i)| =a and |R(y/i)|=b. If xo, yo are a pair 
of any real values of x and y, the function f(z, y) is analytic in the region defined by 
the inequalities |ve—xz)|=a, |y-—yo|=b, and its absolute value is less than MM. 
Then, by the calculus of limits, the integral of the equation y’=/(x, y), which is 
equal to ¥9 for «=%po, is surely analytic in a circle C whose radius r is independent 
of to, Yo. We can follow the analytic extension of that integral along the real axis 
by means of circles of radius r, and we see that it is analytic in the interior of the 
strip bounded by two parallels to the real axis at a distance r from it, 


66 EXISTENCE THEOREMS [II, § 28 


parameter Xin D. These integrals are therefore themselves analytic functions of » 
in the region D (Part I, § 39). 

Most frequently the coefficients az, and 6; are integral functions of the 
parameter \; the integrals are therefore themselves integral functions of X. 
We can obtain directly the developments, according to powers of A, of the 
integrals which take on given initial values, by first substituting in the two 
sides of the equations (47) developments of the form 

Yi = Vio + UA + +++ + UprAP + >>, ((=1, 2,--+, n) 
where the variables u,;, are functions of 7, and by then equating coefficients. 
The functions ujo must take on the given initial values for «= 2,, while the 
other functions uz, where k=1, must be zero for 7 = a. 

Proceeding in this way, we find, step by step, systems of linear differential 
equations for determining these coefficients. We shall return to this subject 
later. 


29. Extension to analytic functions. The method can be extended to complex 
variables. To do so it suffices to observe that we have for analytic functions 
of one or several variables inequalities analogous to the inequalities (40). First, 
let f(x) be an analytic function of a complex variable x, in a region bounded 
by a convex curve ( and also on the boundary, and let A be the maximum 
value of |f’(x)| in this region. The difference f(x,) — f(x,), where x, and @, are 
any two points of that region, is equal to the definite integral [/’(x) dx taken 
along the straight line joining these two points. We have, therefore, 

|F(%_) —F(%)|<A|x2— 2, |. 

Similarly, let f(z, y) be an analytic function of the two variables x and y 
when these variables remain respectively in two regions 2 and 9” bounded by 
two closed convex curves C and C’, and let A and B be the maximum values 


of | f,,| and of |f,| in this region. If a, and a, are any two values of x in Q, and 
y, and y, any two values of y in ”, we can write 


F (®q5 Yo) — fF (®15 Y1) = [F (Los Yo) —F (15 Yo)] + LF (@15 Yo) —F(%, %)]; 


and, consequently, from what we have just shown, we have 


|F (25 V2) —F(®1, ¥y)| <A |e, — @,|+ Bly2— y,|. 
The proof is the same whatever the number of the independent variables. 
Having seen this, let us limit ourselves, for simplicity, to the case of a single 
equation, 


dy 
the right-hand side of which we shall suppose to be analytic in the region defined 
by the inequalities |z— 2|=a,|y—y,)|=b. Let M be the maximum value 
of | f(«, y)| in this region, and h the smaller of the two numbers a and b/M, 


In the plane of the variable z let us describe a circle C, of radius A about the 
point @) as center, and let us put, as above, 


=%t f F(t, Yo) dt, Va =Yot f F(t, Y, (t)) dt, ee) 
0 aa) 


n= Vo +f Flt, irr Ol at, 


II, § 29] SUCCESSIVE APPROXIMATIONS 67 


where the upper limit z is a point within C;,. We prove first, step by step, that 
we have 


Yr — Fal <8, We Vol SO, a — 9G | < 5, 


All the functions 7,, ¥2, +++; Ya, +++ are therefore analytic functions of x in the 
circle C,, and the process can be continued indefinitely. Moreover, we have 


(49) Yn (L) — Yn—1 (x) =f, le Yn—1(t)] —F[t, Yn—2 (t)]} at, 


where the integral is taken along the straight line joining the two points 2), z. 
Let A be the maximum value of |@f/éy| in the region |x — x)|=Sh, |y — yy |=; 
then, according to the observations made just above, we have always 


[7 [t, Yn—1 (€)] —S[E, Yn—2 (|< A] yn-1 (6) — Yn—2 (t) |. 


In order to prove that we have an inequality analogous to the inequalities (45), 
let us suppose that we have 


§£— go, |w—-1 
RN eye Gh a ane elec 


(n—1)! 


which is evidently the case for n= 2. Let x= a, + re”; the change of variable 
t — x) = pe® reduces the integral (49) to an integral taken along the real axis 
from 0 to r, and we have (Part I, § 44) 


r pr-1 yn 
ii oe MAn-1_? dp = MAn-1 
[Yn (2) — ¥ Le\i< f basi ey 
or 
|e 2o|"_ 


ee bo 


The proof can be completed as before. The series whose general term is 
Yn—Yn—1 is uniformly convergent in the circle C,, and, since all the terms 
are analytic functions, the sum of that series is an analytic function in the 
same circle (Part I, §39), which satisfies the equation (48) and which takes 
on the value y, forz=a,. The development in power series of this integral 
is necessarily identical with that furnished by the calculus of limits, but the 
limit obtained for the radius of convergence is greater than that given by the 
first method. 

The remark relative to linear equations applies also to analytic functions. 
Let us suppose that the coefficients aj, and 6; of the equations (47) are analytic 
functions of the complex variable xz. Let us mark in the plane the singular 
points of these functions, and let us suppose that from each of these singular 
points a ray is drawn following the prolongation of the segments from z, to the 
singular point. The set of points of the plane which are not situated upon any 
of the preceding lines is called the star corresponding to the system of singular 
points. The straight line which joins the point z) to a point x of the star does 
not pass through any of the singular points, and the method of § 28 proves 
that all the integrals of the system (47) are analytic functions along that straight 
line. The point x being any point of the star, it follows that all the integrals of 
the linear system (47) are analytic functions in the whole star —a result which 
will be established later in another manner (§ 37). 

The method of successive approximations enables us also to obtain for the 
integrals deyelopments in series converging in the whole star. Let A be a region 


68 EXISTENCE THEOREMS [II, § 29 


of the plane bounded by a closed curve C lying entirely in the star; the series 
furnished by the method of successive approximations are uniformly convergent 
in A. The remaining details of the proof are left to the reader, since they do 
not differ essentially from the details of the proof given before. 


30. The Cauchy-Lipschitz method. The first proof given by Cauchy of the 
existence of integrals of a system of differential equations has been preserved 
in the lectures by Moigno published in 1844. It was considerably simplified by 
Lipschitz, who made clear just what hypotheses were necessary for the validity 
of the proof. 

In order to gain a clear grasp of the whole process, let us take the simple 
equation 


We have shown (I, § 78, 2d ed. ;. § 76, 1st ed.) that the integral of this equa- 
tion which takes on the value y, for = @, is the limit of the sum 


(50) ¥y +S (©o) (@, — Lo) + F (4) (2 — ©) +++ + £(Ln-1) (© — Ln-1), 


where &,, %, +++, %m+1 are. n— 1 points of the interval (x), x), as the number n 
becomes infinite in such a way that all the intervals (x; — 2;_1) approach zero. 
It is this process, suitably generalized, which leads to Cauchy’s first method. 
In order to simplify the exposition, we shall take the case of a single equation, 


(51) oi EE Te): 


We shall suppose that the function f(z, y) of the real variables z, y is continuous 
when «# varies from x, to 2 + a@ and when y varies from Yo —b toy, +6, and 
that there exists a positive number K such that - 


(52) Fe, WY) —F(e, y)|<Kly —y|, 
where y and y’.are any two numbers included between y, — b and Yo + 6, and 
where « lies between x, and a, + a. 

This condition, the importance of which was brought out. by Lipschitz, will 
be called, for brevity, the Lipschitz condition. It has already been used in the 
method of successive approximations (§ 27; and I, § 34, 2d ed. ; § 25 ftn., Ist ed.). 

Let M be the upper limit of | f(x, y)| in the preceding region, and / the smaller 
of the two numbers a and b/M (we suppose a>0,6b>0). In order to prove that 
the equation (51) has an integral which takes on the value y, for x =a, and 
which is continuous in the interval (x, x) + ), we shall imitate so far as pos- » 
sible the procedure followed in establishing the existence of a primitive function 
for f(x). Let x be a value of the variable belonging to the interval. Let us take 
between x, and x a certain number of intermediate values, @,, 7, +++, €i—1, iy 

,%—1, proceeding in increasing order from @, toz. We shall put succeeaiyae 


(53) ¥1=Y% +f (Xo, Yo) (x, — Zp), Ye=Y,+ F(X; Y,) (t_ — 1), af 
and, in general, | 

(54) Yi = Yi-1 +f (@i-1, Yi—-1) (i — B-1). ((=1, 2,++-,n—1) 

The sum 


Yn = Yo tF (os Yo) (& — Xo) + f(y, Yy) (Cy — X) + - 
(G°) a "+ F (tn - 1 ¥6 21) — 9 1) peal al 


II, § 30] SUCCESSIVE APPROXIMATIONS 69 


presents an evident analogy with the sum (50), to which it reduces when the 
function f(x, y) does not depend upon y. We are thus led to investigate whether 
or not that sum approaches a limit when the number n becomes infinite. We 
shall generalize the question by defining first two sums analogous to the quan- 
tities S and s (I, § 72, 2d ed.; § 71, 1st ed.). 

Let us consider the triangle ABC formed by the straight lines defined by 
the equations 


A=% +h, Y=y,+MU(X— x), Y=Y),— M(X—2,). 


From the way in which we have defined h, the function f(z, 7) is continuous 
when the point (x, y) remains in the interior or on the sides of this triangle, 
and its absolute value is at most equal to M. 

The parallels to the y-axis, X=z,, X=2,,---, X =2z, divide the triangle 
ABC into a certain number of isosceles trapezoids of which the first reduces 


to a triangle. Let M, and m, denote respectively the maximum and mini- 
mum values of f(z, y) in the triangle Ab,c,; then we have —-M=m,<M,=M. 
Through the point A let us draw the straight lines with slopes equal to M, and m, 
meeting the straight line X = z, in two points, P, and p,, whose ordinates are 
respectively Y, = yy + M,(@, — %) and y, = yy + m,(%, — Z,). The letter y, no 
longer denotes the same thing as in the expressions (53) to (55). These points, 
P, and p,, are evidently in the interior of the triangle ABC or on its sides, and 
we have Y,>y,. Through the point P, let us draw the straight line with the 
slope M up to its intersection with the straight line b,c, in @,, and through p, 
let us draw, similarly, the straight line with slope — M up to its intersection q, 


70 EXISTENCE THEOREMS (11, § 30 


with the same straight line b,c,. Let M, and m, be the maximum and minimum 
values of f(x, y) in the trapezoid P, Q,q.p,; the straight line with the slope M, 
drawn through P, meets the straight line b,c, in a point P, whose ordinate is 


Y,= Y,+ M, (x, — %,), 


and the straight line with the slope m, drawn through p, meets 6,c, in a point 
p, With the ordinate y, = y, + m,(%,—«@,). We have evidently Y, >y, and 
Y, — ¥,=Y,— y;, the equality holding only if the function f(x, y) is constant 
in the trapezoid P,Q,.9,P,- This process can be continued. Having obtained two 
points, P;—1 and p;—1, on the straight line c;_16;_1, let us draw through P; _; a 
parallel to AB, and through p;_; a parallel to AC. We thus form an isosceles 
trapezoid P;_1Q;q:pi-1. Let M; be the maximum value of f(z, y) in this trape- 
zoid, and m; the minimum value; the straight line with the slope M; drawn 
through P;_; meets the straight line ¢;b; in a point P;, and the straight line 
with the slope m; drawn through p;_; meets c;b; in a point p;. We thus form 
two broken lines starting from the point A, namely, AP,P,--+ Pi-1Pi--++ Pn, 
or L, and Ap, po-++ Pi—1 Pi-++* Pn, Or 1, ending in the two points P, and pp of 
the straight line XY =a. From the manner in which these two lines were con- 
structed it is evident that they both lie in the triangle ABC, that the line L is 
never below l, and that the distance between these two lines, measured on a 
parallel to the axis Oy, cannot diminish when the abscissa increases from @, to z. 
The ordinates Y, and y, of the two extreme points are entirely analogous to the 
sums S and s (I, § 72, 2d ed.;.§ 71, Ist ed.). We shall put S= Y,, 3 = y,. 

To each method of subdivision of the interval (x), x) corresponds a sum S 
and a sum s. If we subdivide each of the partial intervals (x;_, x;) into still 
smaller intervals in an arbitrary manner, the preceding geometric construction 
shows immediately that the line L’ corresponding to this new division is never 
above ZL, and the line U’ is never below I. We have, therefore, S’=S, s’=s, 
where the accented letters denote the sums relative to the second division. We 
conclude from this (as in § 72, 2d ed.; § 71, Ist ed.) that if S, s, S,, s, represent 
respectively the sums relative to any two methods of division whatever of the 
interval (x), x), we have s=S,, s, =S. Indicating by J the lower limit of the 
sums S, and by I’ the upper limit of the sums s, we have, therefore, ’ = I. 

In order that the sums S and s shall have a common limit when the maximum 
length of the partial intervals approaches zero, it is necessary and sufficient that 
S — s approach zero. In fact, we may write 


Bits ig ee ee Te, 


and the difference S — s cannot be less than a number e unless each of the num- 
bers S — I, I— I’, I’ — s (no one of which can be negative) is itself less than e. 
Since e is an arbitrary positive number, this cannot happen unless we have 
I’ =T, and it is, moreover, necessary that S and s shall have the same limit J. 
In order to prove that S — s has zero for its limit, it is not sufficient to suppose 
that the function f(z, y) is continuous, and it is here that the Lipschitz condition 
plays a part. 

Let Y; and y; be the ordinates of the points P; and p;, and 6; the differ- 
ence Y;— y. Since the function f(z, y) is continuous in the triangle ABC, 
corresponding to every positive number \ we can find another positive number 


os such that 
| f (a, y) — f(r’, y’)| = r, 


Il, § 30] SUCCESSIVE APPROXIMATIONS re’ 


provided that the distance between the two points (x, y) and (a’, y’) of the tri- 
angle ABC is less than ¢. We shall suppose that all the differences 7; — x;_4 
are less than ¢. From the construction by which the points P;, p; are obtained 
from the points P;_1, pi-1, we have 


6; = O;-1 + (Mi — mi) (%i — X-1). 


On the other hand, we can write 


Mi — m =f (%;, Yj) —F (7, 97) 
=f (%;, vj) —S(@e, v) + Ov) —FOL YD), 


where (a;, 7/;) and (x;’, y/) are the codrdinates of two points of the trapezoid 
Pi-1QiqGipi-1. We have, therefore, by the condition (52), 


M;—m<r»>++ Klyf—y;|- 


But the difference |v,’ — y;| is at most equal to 6;_1 + 2 M(a;— 2;_}), and we 
have 
Mi—m<dX+ 2 MK (a; — 2-1) + K6;_}. 


If we take all the intervals so small that each of the products 2 MK (x;— x;_,) 
is less than A, the difference M;— m; will be less than 2 ue K6;—1, and conse- 
quently we shall have the inequality 


(56) Oo; < 0;-1 [1 + K (X¢ — eet) + 2r(a;— Pe a8)s 


which can be written in the form 


2X 
+ SB < (5 ae <) + Ker ay Nr 


We have, therefore, a fortiori, 
5; re = =< eR (3 tit: =): 


Putting i=1, 2,---, nm successively in this last inequality and multiplying the 
two sides of the inequalities obtained, we find 


2 2 ‘ 
on + =: = - Gh Fy), 


or 
2X 
S —§ = On <e we [eX @— a») a Lie 


Since it is possible to take the positive number \ as small as we wish, provided 
that all the partial intervals are themselves less than another suitably chosen 
positive number, we see that the sums S and s have the same limit. That limit is 
a function of 2, say F(x), defined in the interval (x), 2) + h). We shall now show 
that this function F(x) is an integral of the given equation (61), and that it 
reduces to y, for =). In showing this we shall continue to make use of the 
geometric representation, 

Tf all the partial intervals approach zero, not only the extremities of the two 
broken lines Z and / approach a limit point, but the lines themselves approach 
a limiting curve. Any straight line parallel to BC meets the line L in a point P, 
and the line / in a point p, and the distance Pp is less than S—s. From the 
properties of these broken lines, all the points P have their ordinates greater 


12 EXISTENCE THEOREMS [II, § 30 


than the ordinates of the corresponding points p; and since the distance Pp 
approaches zero, it follows that the points P and p approach a single limit point + 
lying on the line considered. The locus of these points, zr, is evidently a curve C 
lying between the two broken lines Z and / and passing through the point A. 
The ordinate of a point of that curve with the abscissa z is equal to the func- 
tion F(x) just defined, for in order to obtain the position of the point 7 on the 
line X = a, we make use of only the portions of the broken lines which are on 
the left of that line. Let us suppose the two broken lines LZ and J produced up 
to the side BC, all the partial intervals being less than the smaller of the two 
numbers a, \/(2 MK), and let P(x) and Q(x) be two continuous functions which 
represent the ordinates of a point of the line Z and of the line / in the interval 
(to, % +h). The difference P(x) — Q(a) is less than 2) (e4* — 1)/K, and each 
of the functions P (x), Q(z) differs from F(x) by astill smaller quantity. Since d» 
can be made as small as we wish, we see that we can construct a uniformly 
convergent series of continuous functions in the interval (x), 2) + A) which has 
F(x) for its sum; this function is therefore itself continuous (see Vol. I, § 31, 
2d ed.; § 178, Ist ed.). 

Every broken line included between L and 1 has evidently the same curve C 
for its limit. Such would be the broken line A, whose successive vertices have 
the codrdinates obtained by the recurrent formula 

Ze = M1 + S(@i-1, 2-1) (Li — Vi-1), 
the first vertex being the point (2), y)). Thus we find again the expressions (54) 
which served as our starting point. Let us notice also that if we apply the 
construction starting with a point M’(x’, y’) on the curve C, we obtain two 
broken lines L’ and /’ lying between LZ and l, which also approach more and 
more the portion of C included between M’ and the straight line BC. Let now 
M’ (x, y’) and M’ (x”, y’”’) be two neighboring points of C(2” >a’). The slope 
of the straight line M’M” lies between the maximum and minimum values of 
F(x, y) When the point (x, y) moves over the triangle formed by the straight lines 


Xse",° Y-y=M(4%-v), Yoy =4+ M(x—2); 


if the difference 7” — 2’ is less than a suitably chosen positive number, these 
two values of f(x, y) will differ from f(a’, y’) and from f(x”, y’’) by as little as 
we wish. If one of the two points, M” for example, approaches the first one as 
a limit, the slope of M’M” will therefore have for its limit f(x’, y’). The func- 
tion F(x) consequently satisfies the given differential equation (51). It is, more- 
over, evident that the curve C passes through the point A, that is, that we have 
F(X) = Yo: 

The curve C is the only solution of the problem. If there existed a second 
solution C’, this curve C’ could not be at the same time below all the lines L 
and above all the lines J, since these lines approach the curve C. We can there- 
fore find a line —for example, Z — which will be cut by this curve C’. Since 0’ 
is below the line Z in the neighborhood of the point A, let us suppose that it 
passes above L, crossing that line in a point n; of the side P,_1 P;, and let m;_, 
be the point of C’ with the abscissa z;_1. The slope of the chord m;_;n; is equal, 
by the law of the mean, to the value of the function f(z, y) at a point of the arc 
m;—1 1%; hence this slope cannot be greater than the slope of the side P;_;P;, 
since the arc m,_1”; is in the trapezoid P;_1Q;q:p;-1. But the figure shows 
that the slope of the chord must be the greater. 


II, § 30] SUCCESSIVE APPROXIMATIONS 73 


Cauchy’s first method and that of the successive approximations give, as 
we see, the same limit for the interval in which the integral surely exists. But 
from a theoretical point of view Cauchy’s method is unquestionably superior : 
we shall show, in fact, that this method enables us to find the integral in every 
finite interval in which the integral is continuous. More precisely, let us sup- 
pose that the equation (51) has an integral y = F(x) continuous in the interval 
(9, Z) + J), that the function f(x, y) is itself continuous in the region (£) of the 
zy-plane bounded by the two straight lines «= 2), =a)+ 1 and by the two 
curves Y = F(x) + n, where 7 is a positive number taken at pleasure, and that 
J (x, y) satisfies the condition (52) in this region. Let us suppose that we divide 
the interval (x), ) + 1) into smaller partial intervals and that we construct 
the broken line A by the method which has just been explained, relative to 
this manner of division and starting from the point (29, Yo). If all the partial 
intervals are less than a suitable positive number o, this broken line will lie entirely 
in the region (E), and the difference of the ordinates of two points having the same 
abscissa, taken on the integral curve C and onthe line A, will be less than any positive 
number e given in advance. 

Let £9, £1, Ly, ++°, Li-1, Li+++, Ln—1, Ly + U be the abscissas of the points of 
division, let yp, y,;,---, Y be the corresponding ordinates of the curve C, and let 
Yor 21) 29 °**s Ziy***%y Zn be the ordinates of the vertices of the line A. Let us 
first suppose that all the vertices to the left of the vertex (a, z,;) are in the 
region (#), and let us consider the problem of calculating an upper bound 
for the difference d; =|z;— y;|. 

We have, on the one hand, from the very definition of A, 


Pe eta + (i=, 22-1) (4s — Fi —1). 


On the other hand, from the law of the mean, we have also 
Yi = Yi-1 +S (@j, Y;) (Gi — Bi-1), 


where (x;, y;) are the codrdinates of a point of C, and where 2; lies between 
a;—, and z;. We derive from these equations 


(57) 2— Ys = Zi—-1 — Yi-1 + (i — Vi-1) [F (@i-1, 2-1) —S (@ YI} 
and the coefficient of (x; — 2;~1) can be written in the form 


[Sf (ai—1, 21-1) —S (@i-1, Yi-1)] + [fF (@i-1, Yi-1) —F (2 YI. 


The absolute value of the first difference is, by the condition (52), less than 
Kd;_1. On the other hand, since the function f(a, y) is continuous in the region 
(£), it is a continuous function of ¢ along C, and we can find a positive number ¢ 
so small that | f(a, y) — f(#’, y’)| is less than a given positive number 2 ) for any 
two points of the curve C, provided that |x — a’| is less than «. Having chosen 
the number o in this way, we have 


(58) di < di-1 + (ti— t%~-1) (2X + Kaj-1), 


° 
a relation which is very similar to the relation (56), and from which we obtain, 


as before, the inequality an 
a; <= fexe—a — 1), 


T4 EXISTENCE THEOREMS [1I, § 30 


Let us suppose that the number ) is so small that we have 2) (eX! —1)< Kn. 
We may then establish, step by step, that each of the differences d,, d,, +--+, dy is 
less than 7. All the vertices of the broken line A are therefore in the region (£). 
Let P(x) be the ordinate of a point of the line A; similarly, let Q(x) be the 
ordinate of a point of the auxiliary broken line A’ obtained by joining the 
points of C having the abscissas @, 2, %),+++, In—1, %) + /. Then we have 


P(x) — F(z) = P(z)— Q() + Q(@) — F@). 


If the oscillation of the function F(z) in each of the partial intervals is less 
than ¢/2, we have always | Q(z) — F(z)| <«/2 (see Vol. I, § 206, 2d ed.; § 199, 
Ist ed.). If also the number 7 is less than e/2, we have | P(x) — Q(x)|<e/2, and 
therefore | P(x) — F(z)|<e. Then the continuous function P(x) represents the 
function F(a) with an error less than e in the whole interval (x), + ). 

The Cauchy-Lipschitz method can be extended to systems of differential 
equations without any other difficulty than some complications in the formule. 
It applies also to complex variables. The investigations of E. Picard and of 
Painlevé have shown that the method leads to developments of the integrals in 
convergent series in the whole region of their existence if the right-hand sides 
of the given equations remain analytic in this region. 


Ill. FIRST INTEGRALS. MULTIPLIERS 


31. First integrals. Given a system of n —1 analytic differential 
equations of the first order, we shall write these equations in the 
symmetric form 


(59) SSS ee Se 


where the denominators X,, X,---, X, are functions of the nm variables 
4) Ly,+++, %,- This form of the equations does not involve a choice 
of the independent variable, which may be any one of the variables 
or may be chosen arbitrarily. We have seen above that, under 
certain conditions which have been defined, all the integrals of this 
system which pass through any point of a region D are roprese as 
by a system of equations of the form 


(60) {7 Wy, +++, I) = Cy, Fy (By Lay 20 2y a) = Coy "aneag 
Fa-1@y Hoy 22 %y Xn) = Cy) 

where f\, fy +++) f,-1 are (n — 1) functions analytic in D, and where 
Cy, Coy + + +) Cy—1 are constants which may be arbitrarily chosen, at least 
within certain limits (§ 26). The formule (60) represent the general 
integral of the system (59) in the region D; but there may be other 
values of the variables also, for which (60) represents the solution. 
It may happen that we obtain several different systems of formule 
representing the general integral in different regions. It is also clear 


II, § 31] FIRST INTEGRALS. MULTIPLIERS T5 


that, in the same region D, the system of equations (60) is not the 
only possible representation. We can replace the (m —1) functions 
F; by (n —1) functions F; which depend only upon the functions f,, 
provided that these (n — 1) functions F; are independent functions 
of the variables f,. 

However the functions /; have been taken, if the formule (60) 
represent the general integral of the system (59), the functions /f, 
satisfy the same partial differential equation of the first order. For, 
let us suppose the codrdinates of a point 2, #,,---, x, of an integral 
curve expressed as functions of a variable parameter. If we replace 
the codrdinates x,, x,,---+, #, in f; by their expressions as functions 
of this parameter, the result reduces to a constant. We have, there- 
fore, df; = 0, and, replacing the differentials dx,, dx,,--- in df, by 
the proportional quantities X,, X,,---, we find that f; satisfies the 
relation 


(61) X(f)= Ey ee Bet tig =o 


n 


This relation must reduce to an identity, when f is replaced by jf, 
since we can choose the constants C; in such a way that the integral 
curve passes through any point of D. The (m —1) functions f, f, 
+++, f,-1 are therefore (xn —1) integrals of the equation X(f)= 0 
Every function II(f,, f,°-+,fn-1) 18 also an integral of the same 
equation, whatever may be the function II, by the relation 


XQ) = FXG) + Gp A +o tae 


which is easily re | 
Conversely, we obtain in this way all the integrals of the equation 
X(f)=0. For, eliminating the coefficients X,; from the n relations 


X(f) = 9, X(f,) = 9, TA “Ue — 0) 
we obtain 


ye oe pela DE ty 
D (4, Lay ++ +) Ln) 


which shows that fis a function II (f,, f,, --+;f,—1) of the (n —1) par- 
ticular integrals f,, f,,:-+, fr-1 C1, § 55, 2d ed.; § 28, 1st ed.). We 
can also verify this by a change of variables. Let us suppose, in fact, 
that we take a new system of independent variables y,, y,, +++; Yay 
where the n — 1 variables y,, y,, +--+, Y,_1 are precisely the functions 
Fy) too ++ *> Jn—1 themselves, and where the variable y, is chosen in 
such a way as to form with y,, y,, +--+, Y,-1a System of m independent 


76 EXISTENCE THEOREMS [II, § 31 


functions of the original variables x,, 7,,--+-+,2,. Then the equation 
X(f) = 0 is replaced by an equation of the same form 


of sf = 
(C2 NE NNO = Verret =0, 
which must have the a os 1) seas Pen 
S= Yp Ske) f= Yn—1° 


We have, therefore, 


ay: 


Y=HY,=::°= eet 


and the equation (62) reduces to éf/éy, = 0. The general integral is 
therefore an arbitrary function of y,, y,, +++; Yn—1-™ 

The integration of the partial differential equation X(f) = 0 is 
therefore reduced to the integration of the proposed system of dif- 
ferential equations (59). Conversely, let us suppose that we have 
obtained an integral f of the equation X(f)=0 in any manner 
whatever. If we replace z,, x,,---, x, in that function by the coér- 
dinates of a point of an integral curve, supposed to be expressed as 
functions of a variable parameter which may be one of the coordi- 
nates themselves, the result obtained reduces to a constant. In fact, if 
we suppose that x,, x,,---, #, are functions of a variable parameter 
satisfying the relations (59), the total differential df of the preced- 
ing function reduces to KX(f), where K denotes the common value 
of the ratios dx,;/X,. The equation f= C is therefore a consequence 
of the given system of differential equations. For this reason we say 
that the function f is a first integral of that system.t 

If we know n —1 independent first integrals, we can write im- 
mediately the general integral of the system (59) ; if we know only 
p independent first integrals (p < n —1), we can reduce the integra- 
tion of the given system to the integration of a system of n — p —1 
differential equations. For, let f, f,, ---, jf, be these p first integrals. 
From the p relations 


J, =Cy J, = Cy nh? Sr = Cp 


* The two modes of reasoning do not require that the function f should be analytic. 
The only necessary conditions are those which are required in order that we may 
apply the formule for change of variables, that is, the existence and the continuity of 
the partial derivatives of the desired function /. 

+ The reasoning would no longer apply if the factor K were infinite for all the 
points of the integral curve, which would be the case if the coordinates of all the 
points of that curve were to make the n functions Xi vanish. It is also necessary 
to make an exception of the integrals which are such that at least one of the functions 
X,, X2, +++, Xn is not analytic in the neighborhood of any point of that curve. This 
case arises when there are singular integrals. 


II, § 31] FIRST INTEGRALS. MULTIPLIERS (fi 


we can obtain p of the variables ,, ~,,---, x,, for example, «,, a,, 

--, x, aS functions of the remaining » — p variables 2,,,, ---, 
x, and the p arbitrary constants C,, C,,--+-, C,. It will suffice, then, 
to determine «43, T42)°**) Z,_ a8 functions of x single independent 
variable. If we denote by X,.4,, Xp,42,---+, X, the new functions 
resulting from X, 41, X,+2,°:+,X, after we have replaced ~,, x,, -- -, 
x, in them by their expressions, it will suffice, therefore, to integrate 
the new system, 


(63) Uy +1 = diy +3 —— ae Lay 
X41 Xp +2 xX, 


in which the new denominators depend upon p arbitrary constants. 

We can also reason in another way. If we take a new system of 
independent variables, ¥,, ¥,, +++; Y,, Where the p variables y,, y,,---, 
y, are identical with the » known first integrals f, f,,---, j,, the 
equation X(f)=0 is replaced by an equation of the same form, 
Y(f)= 090, which must have for integrals f=y,,---, f=y,. That 
equation is therefore of the form 


é 0 

AES f+ .6. 4 ie a = 0, 

OY» toe OYn 

and its integration reduces to that of a system of n — p —1 differ- 
ential equations of the first order, 


pti 


We see from this the importance of looking for first integrals. 
In each particular case the discovery of a new first integral con- 
stitutes a step farther toward the complete solution. It would not 
be possible to give a very definite rule of procedure for this purpose. 
Let us merely notice that the problem amounts to forming an inte- 
grable combination of the equations (59), that is, to determining n 
factors, {,, Mg) ** +) Mn, So that 

BX, bp X, +> + HX, = O, 
and that 
fda, + pda, +--+ + p,dx, 


is an exact differential dd. For it is clear that we can deduce from 
the equations (59) a new ratio equal to the first 


dx; ‘Wer fy da, Re eda ot Hn Abn 
X; Petits? te bg Xe, 
hence the relation 
do = Lda, +.--+y,dx, =9 


78 EXISTENCE THEOREMS [II, § 31 


is a consequence of the equations (59) if 

pp pine + p,X, = 0. 
It follows that we can find a first integral by quadratures if we 
know the factors yw; This is the case in particular whenever we 


can find n factors, “,, @,,+-*; M,, Such that the factor w; depends 
only upon the variable x;, and such that 


au, X; = 0. 


Let us also observe that, if we have obtained p first integrals of 
the system (59), it may happen that the new system (63) can be 
integrated completely for particular numerical values of the con- 
stants C,, C,,--+-+, Cp, while the actual integration is impossible for 
arbitrary values of these constants. 


Example 1. Let it be required to integrate the system 


(64) ae = vw, ia = wu, ase 
dx dx dx 


= Uv. 


We easily see two integrable combinations udu = vdv = wdw. We have, there- 
fore, two first integrals, u? — v? = C,, u2?— w* = C,. Hence, putting the values 
of v and of w obtained from these relations in the first of the equations (64), 
we have for the determination of u the differential equation 


= Vie 6) (B= GC), 
the general integral of which is an elliptic function (§ 11), reducing in special 
cases to a simply periodic function or even to a rational function. Since the 
given system is symmetric in u, v, w, we conclude that v and w are also elliptic 
functions. 

Example 2. Let us consider the system. 


(65) 


(66) ad ae ok w ee w— TU a of u v 
obs qu, aah ’ AE et pr, 


where p, q, r are given functions of x. We have again an integrable combination, 
udu + vdv + wdw = 0, from which we derive the first integral, uw? + v? + w? = C. 
Discarding the case where C is zero, we may suppose C = 1, for the system (66) 
is not changed by multiplying u, v, w by the same constant factor. Instead of 
solving the relation u? + v? + w? = 1 for one of the unknowns, we can proceed 
in a more symmetric manner by considering wu, v, w as the codrdinates of a 
point of a sphere of radius unity and expressing them as functions of two varia- 
ble parameters — for example, in terms of the parameters which determine the 
rectilinear generators of the sphere. Let us put for that purpose 


ut te lye uti Ll—w_ 
1—w u—w 1tw u—iv 


— fy 


which gives 


II, § 31] FIRST INTEGRALS. MULTIPLIERS 79 


Substituting these values of u, v, w in the system (66), we find after some 

easy calculations that \ and » must satisfy the same Riccati equation, 
do : q—p gt 

67 —=-—ir ——— 4 —_— o?, 

a) dix ett SN ee 
Hence the integration of the given system is reduced to the integration of a 
Riccati equation.* 

Example 3. Let us consider the equation integrated by Liouville, 


y+ o(e)y +fy)y?=0. 
Putting zy’ = z, we may replace the given equation by the system 


CpG Us vis 1 MaNES 
1 2 b@)zt+fy2 


from which we derive the integrable combination dz/z + ¢(x) dx + f(y) dy = 0. 
The given equation of the second order has therefore the first integral, 


2 y 
es dade fi sua _ C, 
which we could also have obtained directly by dividing all the terms of the 
equation of the second order by y’. The preceding equation of the first order 
is of the form y’= CXY; hence, by separating the variables, the integration 
may be completed by two quadratures. 


Note 1. We sometimes replace the system (59) by the system 


(68) sets SB a. = Ea, 

Pia) 2 n 

where ¢ is an auxiliary variable which is introduced in many cases only for the 
sake of greater symmetry in the reasoning. If the original system (59) has 
been integrated, we can obtain t by a quadrature, for if we replace z,, x3, ---, 
Zn, for example, by their expressions in terms of x, and of the constants C,, 


Cy, +++, Cr—1 in X,, we are led to a relation, 
dt = P(z,, C,, Cz,-+-, Cn—1) dz, 


from which we can find ¢ by a quadrature. It follows from this that the gen- 
eral integral of the new system (68) will be represented by the n equations of 
the form 


69° 
( ) Fn (24; Te, °° %5 In) =t— by, 


‘. = CT; te = Co Bh s=6'4 ant = Coe1: 


where f,, fg, --+, Jn-1 are (n— 1) independent integrals of X(/f)=0, and 
where ¢, is a new arbitrary constant. 

Conversely, in order to obtain the integral curve of the system (59) that 
passes through the given point a, 72,---, «2, we can look for the integrals of 


ee 


* See DarBoux, Théorie des surfaces, Vol. I, chap. ii. 


80 EXISTENCE THEOREMS [II, § 31 


the system (68), where ¢ is considered as the independent variable, which for 
t = 0 take on the values x}, x2,---, x2 respectively. Let 


‘ee Wey 85 tn) Ly = d2a(t; yy tes tp), St 


70 
(70) Ln = Dalby loge sy ae) 


be these integrals ; it is clear that the preceding expressions represent the inte- 
gral curve sought. We should have to make an exception only if all the func- 
tions X; were zero for the initial values 2? and analytic in the neighborhood. 
In this case the expressions (70) should reduce to 2; = z?. But, since the ratios 
dx,/dx,,+++, dt,/dx, appear in an indeterminate form, nothing justifies us so 
far in saying that there is no integral curve passing through the given point. 
This is a case which will be examined later (§ 75). 

Note 2. The relation which exists between the system of differential equa- 
tions (59) and the linear equation (61) proves that X(f) is a covariant of the 
system (59). The meaning of this statement is as follows: Let us suppose that 
we take a new system of independent variables, y,, y,,+++, Yn, connected with 
the variables £,, 22, +++, Zn by the relations 


(71) Li = Pi (Ys Yos***s Yn)- (j= 1, 2,-++, n) 
By the formule for change of variables, @f/dz; is a linear homogeneous func- 
tion of the derivatives @f/éy;, and X(f) changes into an expression of the 
same form, 

of of of 
72 Y (f)\ =e OY ee 

(72) eX (Z) Matin Wai ae 

where Y,, Y.,-++, Yn are functions of y,, ¥.,°++, Yn. This being true, we may 


now assert that the same change of variables applied to the system (59) leads 
to the new system of differential equations, 


0, 


(73) LTE YRS, rel ta E)) 
Yai Y, 


We could establish this by a direct calculation, but it results also from the 
preceding properties. In fact, let 


(74) Shh ra & 
1 2 n 
be the system to which we are led by applying to the original system (59) the 
change of variables (71) ; it suffices to show that Z,, Z,,---, Z, are proportional 
to Y,, Yg,-++, Yn. Now let f(x,, %,--+, %n) be a first integral of the system 
(59) and 
F(y,, Yogi as Yn) 


the function derived from f(z,, %,+++, %n) by the change of variables. Since 
we have X(f) = 0, we have also Y(F)=0. Besides, F(y,, y,,-+-+, Yn) is evi- 
dently a first integral of the new system (74), that is, an integral of the linear 
equation 

oF oF 

Z(F) = Z, — +---+2Z,—=0. 

Oo” OYn 
Since the linear equations Y(F) = 0, Z(F) = 0 have the same integrals, their 
coefficients are proportional, which proves the theorem. 


II, § 32] FIRST INTEGRALS. MULTIPLIERS 81 


This last point in the proof results from the fact that a linear equation 
X (f) = 0 is completely determined, except for a factor, when we know (n — 1) 
independent integrals, f,, fo,°++, fn—1, Of it. In fact, the (n—1) equations, 
X (fj) = 0, linear and homogeneous in _X,, X,,--+-, X,, determine the ratios of 
these coefficients as unknowns, for the determinants of order (n— 1) formed 
from the partial derivatives of the functions f; cannot all be zero at the same 
time (I, § 55, 2d ed.; § 28, Ist ed.). It may be noticed that the most general 
linear equation having the (n — 1) integrals f; can be written in the form 


DS Sah das «2+, Jtn—1) 2G, 


aE at 8) D(x, , & Ln) 
1? Ci same ori n 


? 
where II (x,, %,+-++, 2») is an arbitrary function. 


32, Multipliers. The theory of integrating factors has been extended by 
Jacobi to simultaneous differential equations. Let /,, f,,---, f,—1 be independ- 
ent first integrals of the system (59). The equation X(/) = 0 is, as we have 
already remarked, identical with the equation 


Ar D(fielaaves _ aeta 1) =i yy 


D (1, %y +++, Ln) 


Writing the condition that the coefficients of the derivatives of/éz; in the two 
equations are proportional, we are led to n relations which may be written in 
the form 


(75) . A; = MXi, (i =1, 2,---, n) 


where A; denotes the coefficient of @f/dz; in the determinant A. This factor M 
is called a multiplier. 

Whatever the first integrals f,, 4, --+, :—-1 may be, this function M satisfies 
the linear partial differential equation 


(76) 6(MX,) a 6 (MX,) Ae Ae 6(MX,,) ry 
OL, OL, OLn, 

Substituting for each of the products M_X; = A; its equivalent expression as 
a determinant of order n—1, and carrying out the indicated differentiations, 
each term of the left-hand side is, in fact, the product of a derivative of the 
second order, such as 62/;,/0x; 0x, (i Ak), and (n — 2) partial derivatives of the 
first order. To prove that the result is zero, it suffices to show that it does not 
contain any derivatives of the second order. Let us take, for example, the 
derivative 6?f,/dr,érz,. This derivative appears in two terms; in one it is mul- 
tiplied by D(f,, fg, +++; fr—1)/D (5, &4,+++, Ln), and in the other by the same 
coefficient but with the opposite sign. The sum of these two terms is therefore 
zero, and similarly for all the others. 

If M, is a particular integral of the equation (76), the substitution M= M,u 
reduces that equation to the form X(u) = 0. If we know a multiplier M of the 
system (59), the general integral of the equation (76) is accordingly MII(/,, fy, 
++, fn—1), Where II is an arbitrary function. Every function of this form is 
also a multiplier ; in other words, there exist (n — 1) first integrals F,,+-+-, Fn -1, 
such that MII(f,, /,,°++,; 4-1) can be deduced from F,, F,,--+-, #,-1 in the 


82 EXISTENCE THEOREMS [II, § 32 


same way that M was deduced from f,, f,,---, f,.-1. For this purpose it is 
sufficient that we have, supposing X, < 0, 


nas D(F,, F;, neaias F,, -1) iy * Ae D(F,, F,, ae Fy) D (Fi, Sos +5 Sn—1) — MO 
X, D (Xe, gy +++, Ln) X, D(fis Sos ++ +s fn—1) D(q, +++, Ln) , 


or 
D(F,, | ee Fy, —1) 


Dyas thers ea Sant) 


This condition can be satisfied in an infinite number of ways. Indeed, n— 2 
of the first integrals F; may be assigned arbitrarily in advance. 
Let us consider the system . 
iy a Tg in 
X caek Cae ene 


= IE Chas Sas "f -*,Sn—1)- 


(77) 


with the auxiliary variable ¢. This system can be reduced to the simple form 
(78) dy, = dy, =+-- = dy,-1= 9, dy, = dt 


by taking for the variables the n —1 first integrals f,, f,,-++, f,-1 and the 
function f,, which appears in the preceding formule (69). It is easy to obtain 
the general expression for the multipliers in terms of the variables y;, for 
every multiplier is of the form _ 


1 Dy, Yor t**s Yn—1) 


M=— he Josey MeN 
Yds (ee ey a tua) (Ys Yor ***y Yn—1) 
On the other hand, we have 
Baltes emer Gt, Yn da 
1 dt ~ by, dt Dyn Gt On 


From the relations y, = /,, +--+, Yn =Jn, Which define the change of the variables, 
we derive, by differentiating with respect to y, and solving, 


Dy, Yos ae a) Unowt) 


Oy (10-1 D (x2, LO eS Ln} ’ 
C’n DY; Yo car) Yn) 


D1, Lay > +5 En) 
and the general expression for the multiplier can be written in the form 
* = D (xy, oy +* +s Ln) 
MM. D(Ys, Voxs>*s Yn) 
where @ is an arbitrary function of ¥,, Y%, +++; Yn—1- 
Let us suppose, now, that after carrying out any change of variables affecting 


only the z;’s without changing the variable t, we have reduced the system (77) 
to the form 


(79) (y;, Yor *%s Yu), 


dx; dx. dx’ 
(80) eee eee ne We, 
Xess DE 


where the X;’s are functions of the new variables x; independent of t. If M’ is 
a multiplier of this new system, we have 
1 D(@;, #5, +++, &) 


(81) i CONE ATTy, AY Yor? ?%s Yn—1)- 


1 


II, § 33] FIRST INTEGRALS. MULTIPLIERS. 83 


Taking the same function @ in the two expressions, we derive from them, by 
dividing their corresponding sides, the relation 


(82) = we el 
D(x, Tq) ***y Lp) 


Hence, if we know a multiplier M for the system (77), we can derive from it a 
multiplier M’ for the transformed system. 

This property.explains the practical importance of multipliers. Let us sup- 
pose that we know n — 2 first integrals of the system (59), and also a multiplier. 
We can then reduce this system to the form 


dy nwa _ nn 
t 0 Xr Fn 


by a change of variables, and we can then find a multiplier M’ for this new 
system, that is, a solution of the equation 


o(M’ XxX’ (MX 
( et) se ( 2) 6 


7 


OEE! On, 
It follows that M’ is an integrating factor for X),dx,_,—X,_,dz;Z, and the 


integration can be finished by quadratures. 

A particular case which presents itself frequently in mechanics is the one 
for which we have 26X;/é2; = 0. The equation (76) reduces then to X(M)=0, 
and we know at once a multiplier M=1. 

This remark applies also to the equation of the second order, y” = f(z, y), the 
integration of which leads to that of the system 


ie ee 
Py FH) 
If we know a first integral of it, y(z, y, y’) = C, we can, from what precedes, 


finish the integration by quadratures. This is easily verified as follows: Let us 
suppose that the equation y (x, y, y’) = C has been solved for y’: 


y = o(2, y, C). 


Since all the integrals of this equation of the first order must satisfy the equation 
y’ =f (x,y), whatever may be the constant C, we must have 6¢/6x + (6¢/dy)¢=/f. 
Hence, since f does not contain C, 


ao an) Op o _ 4 
Cou  OCoby oy 6G * 


which states that 0¢/0C is an integrating factor for dy — ¢dz. 


33. Invariant integrals. The invariant property of the multipliers relative to 
every change of variables can be brought into relation with the general theory 
of invariant integrals, due to Poincaré,* and about which we shall say a few 


* Les méthodes nouvelles de la Mécanique céleste, Vol. III, chap. xxii, and the 
following chapters. See also GoursAT, Sur les invariants intégrauz, in Journal 
de Mathematiques, 6th series, Vol. IV. 


84 7 EXISTENCE THEOREMS [II, § 33 


words. Let us consider in particular a system of three differential equations, 


(83) Se eee 
SOM, AY AZ: 

where X, Y, Z are functions of z, y, z. In order to simplify the statements, we 
shall regard these equations as defining the movement of a particle in space, 
where the variable ¢ represents the time. The particle which, at the time ¢ = 0, 
is at a point M, (Xo, Yo; 29) has arrived at the time ¢ at a point M; whose coodrdi- 
nates are (x, y, z). If the point M, describes a certain region D, of space, the 
point M; describes a corresponding region D;. Now let M(a, y, z) be a function 
of the variables x, y, 2; we shall say that the triple integral 


r= {ff me, y,.2) dx dy dz 


is an invariant integral of the system (83) if the value of that triple integral, 


iE if [M (2, u, 2) de dy de, 


extended over the region D;, is independent of ¢ and equal to the same inte- 
gral extended over the region D,). For example, if the equations (88) define 
the movement of an incompressible fluid, the volume of the region D; is constant 
and the integral {ff dxedydz is an invariant integral. 

Invariant line and surface integrals are defined in a similar way. If the 
point M, describes a curve L, or a surface 3%, the point M; describes a curve 
LT; or a surface 3. A line integral 


fade + Bdy + ydz 


is an invariant integral if the value of that integral along the curve L; is inde- 
pendent of t and equal to the same line integral taken along Ly. Similarly, a 
surface integral 


fe Pdydz + Qdzda + Rdxdy 


is an invariant integral if the value of that integral extended over the surface 3, 
is independent of ¢. 

These notions can be extended without difficulty to the most general systems 
of differential equations of the form (68). For such a system there are n classes 
of invariant integrals, of the 1st order, of the 2d order,---, of the nth order, 
according to the order of multiplicity of the integral considered. The conditions 
that a multiple integral of order p shall be an invariant integral are easily ob- 
tained by means of the formule for the change of variables in multiple integrals. 
We shall develop the calculations for a multiple integral of order n. Let 


TO) = fi fi--+ fi Mey, ta, ++ +5 ty) dey dy ++» diy 


be a multiple integral of order n extended over the region D, which corresponds 
to a definite region D, in the manner just explained. This integral will be an 
invariant integral if itis independent of ¢ ; that is, if we have I’ (t) = 0. In order 


II, § 33] FIRST INTEGRALS. MULTIPLIERS 85 


to calculate that derivative, we shall give to t an increment hf, and we shall cal- 
culate the coefficient of h in the development of I(t + h). Let x; be the value to 
which 2; changes when we change t tot + h; we have 


(t+ mr) = fff) Mei way +25 0) day ding +++ dey, 
t 


where the new integral is extended over the region D,, which corresponds point 
for point to D;. Then we may write 


De, Lo, com ce 9 «;,) 
Ti t h eas eee M if eee rf eS oP eA sy The ‘ele 6 n! 
( “+ ) {ao ie (x, ’ Ln) D(z,, Le, eon Ln) dx, dx, ALn, 


On the other hand, omitting the terms in h of degree higher than the first, 


we have 
w= t+ hXp+ ++, 


yd Vg / oM oM 
M (21, ag, °° *, £,) = M (21, Ly, °*+5 Ln) + n(x, + eit Ae )+ vee, 
Ly Lea 
Panes RY a eS 
D (ai, 3 +++ 29) ay ne sk 
= DG 
D (21, Lg, +++; Ln) jue | hay cae 
Ly 0X5 


=14+n(=34... 42) 
Ox, OLn 
and 
7a ey: , D(z; Tigy vi, a) 
M(x, Lins * * (a so a es 


D(x, Lo, ee) In) 


ox, OXn 0M é 
+ nf a(t + Jag tet] tee. 


= M(x, &, +++, Ln) 


cry 


The derivative dZ/dt has therefore the value 


dig a(MX,) (MX) 
“oi Gaal On, fee SO) te ty +d, 


In order that J be an invariant integral, it is necessary and sufficient that dI/dt 
be identically zero, whatever may be the region D, and therefore that we have 


6(MX,) O(MXn) _ 
erry DO a as is in 0. 


(84) 


This condition is identical with the equation (76), and we obtain Poincaré’s 
theorem: In order that the multiple integral 


Sf faldey +++ den 


shall be an invariant integral, it is necessary and sufficient that M be a multiplier. 
It follows that if we make any change of variables, 


Li = Pi(Y1, Yort**s Yn)s (ij=1, a n) 


86 EXISTENCE THEOREMS [II, § 33 
in the equations (77), we obtain a new system, 


(77’) 


eee 


and if M is a multiplier of the system (77), the n-fold integral 


[fof Mae, deg «+ dey 


is an invariant integral of that system, and the n-fold integral which is obtained 
from it by the same transformation, 


TD (Pagekigott tty Sn) 
Bree 0 Bt Gove Cos Ty) dy dy --+ dy 
Sf J (1, : D(Yy5 You + **y Yn) 12 ck 


is evidently also an invariant integral of the transformed system (77’). Therefore 


the expression 
D(&1, &, +++, Ln) 


Dy, Yor ***s Yn) 


Moa iM 


is a multiplier of the new equations (77’), as we have demonstrated directly. 


Example. In order that the volume shall be an invariant integral of the 
equations (83), M=1 must be a multiplier, which requires that we have 


0X), Y: a2, 


(85) ‘bx oye) eee 


This is the condition for the incompressibility of a fluid for which the equa- 
tions (83) define a stationary flow. 


IV. INFINITESIMAL TRANSFORMATIONS 


84. One-parameter groups.* Every set of an infinite number of transformations, 
of any nature whatever, affecting the n variables z,, 7,-+++, %, form a group if 
the transformation obtained by carrying out any two transformations of this set 
in succession belongs to the set. For definiteness let us consider two variables 
x, y, and let T be the transformation defined by the equations 


(86) v=f(t,y; 4), y=o(t,y; 4), 


where a denotes an arbitrary parameter. If we regard x and y as the coordi- 
nates of a point M in a plane, and 2’ and y’ as the codrdinates of another point M’, 
the preceding equations define a point transformation. To each value of the 
parameter a corresponds thus a definite transformation. Varying this param- 
eter, we obtain an infinite number of different transformations. Let us suppose 
that we carry out in succession two different transformations of this set, corre- 
sponding to any two values a and 6 of the parameter. The first transformation 
will carry the pair of values (z, y) over into the pair of values (a’, y’) given 


* The theory of continuous groups of transformations was developed by Sophus Lie 
in a great number of papers and in his treatise, Theorie der Transformationgruppen. 


II, § 34] INFINITESIMAL TRANSFORMATIONS 87 


by the equations (86). The second transformation will then carry the pair of 
values (x’, y’) over into a third pair (x”, y”) such that we have 


(87) a=fir',y; 6), y=ol(r,y’; 0). 


Let us replace 2’ and 2 in these last two equations by their values (86). The 
resulting equations, 


(88) AN Gh RL at) Ye Py de Os 


again define a point transformation depending upon the two parameters a and b. 
We shall say that the set of transformations (86) form a continuous one-parameter 
group if the new transformation (88) belongs to this set. It is necessary and 
sufficient for this that the equations (88) be of the form 


(89) gv” =f (x, Y; C), y= p(x, Y; c), 
where c is a value of the parameter depending only upon a and upon 6; that is, 
c=y/(a, b). The preceding definition evidently applies whatever may be the 
number of variables, in particular if there is only a single variable. 

The relation z’ = x + a, or, any one of the pairs of relations 


o = 2+ a, y=yt2a; 
xv =xcosa— ysina, y =xsina+ycosa; 
HAGE mids i yo ary 


represents a one-parameter group. On the contrary, the transformations z’=2+a, 
y =y +a? do not form a group, for the transformation resulting from two suc- 
cessive transformations, 7” =x+a+b,y” =y+ a* + b?, do not belong to the set. 

If in the equations (86), which define a group of transformations, we put 
a= II(a), where @ is a new parameter, it is clear that the relations obtained 
again define a group. The same thing is true also if we make a change of vari- 
ables, as we easily convince ourselves a priori. In fact, if a set of point trans- 
formations in a plane is such that the transformation resulting from two 
successive transformations belongs to the set, it is clear that this property is 
independent of the choice of the codrdinates by means of which we fix the 
position of a point in the plane. It is easy to verify this directly. Let us 
suppose that we put x= II(u, v), y= T1,(u, v), and let the inverse relations 
be u= H(z, y), v= Ty} (2, y), so that we have identically 


Bawa (ec, etl (ey), y= [Me y), Dy te, y)]. 


By hypothesis, the transformations considered form a group, and the equa- 
tions (89), where c=y(a, b), are a consequence of the equations (86) and (87). 
Let (u, v), (u’, v’), (w’, ¥”’) be the pairs of values of the new variables which 
correspond respectively to the pairs (a, y), (x’, y’), (2, y”). We have 


uv TI *(2’, y)= i? { FCI (u, v), I, (u,v); a], ¢ [II (u, v), Il, (u, v); a}} 
= Fu, v; a), 

¥ =Tyt@,y) =U; {fs (y, »), 1, (u, »); a], o[1(u, v), Hy (u, v); @]} 
= (u,v; a); 


(90) 


and everything depends on showing that the equations (90) also define a group 
of transformations. Now we have, for example, u” = F'(u’, v’; 6), or 


uw” = I *{ Ff [M1 (wv, v’), Wy (wv, v’); 6), @[ Mw, v’), I (w’, v’); b]}. 


88 EXISTENCE THEOREMS [II, § 34 


Since the equations (86) define a group, this value of uw” is equal to 
Were fa ear y’; 6), d(’, y’; 4)) = Use Ce, Y; C), d(x, ¥; ¢)]; 
that is, to 
A ee (u, v), I(u, v); ¢], [I (u, v), I, (u, v); c]}= F(u, v; c). 


Similarly, we should find that v’ = @(u, v; c). Two groups of transformations 
which are carried over one into the other by a change of variables are said to 
be similar. For example, the two groups 2 = az, u’ = u-+ 0 are similar, for we 
pass from one to the other by putting u = loga, b = loga. 

We shall now determine all possible one-parameter groups, supposing that 
the functions fand ¢ are analytic, and supposing also that the group contains the 
identical transformation, that is, that for a particular value a, of the parameter 
we have f(@, ¥; d)) =X, P(X, y; 4) = y, Whatever x and y may be. 

In the equations of condition 


(91) f(y’; )=Sf(t, ¥3; °), g(x,y; 6) = (2, Y; ©) 
we can consider gz, y, a, c as independent variables, and b as a function of a 
and c defined by the relation c=y(a, 6); x and y’ are functions of 2, y, 


and a defined by the equations (86). Taking derivatives with respect to a, we 
derive from the relations (91) 


of oe , of df fh _4 29 ae , 09 dy , 06 & 


= — —-—0. 
Ox’ Ga | er da «db da =Stié«’si“‘<‘«é‘“é’' —=é«é‘’' G—~=—«sOD CTL 


(92) 


But 6b/éa is given by the relation dy/da + (éy/db) (6b/da) = 0, and therefore 
depends only upon a and b. Solving the preceding equations (92) for dx%’/da, 
dy’/da, we obtain, therefore, relations of the form 


Ox’ ve / é , / / 
SMa DEG, 9, )) — S- = A(a, 0) nv, 0). 


Now z’ and y’ do not depend upon 0b; the same thing is therefore true of \, ¢, 7 
if they have been properly chosen. Therefore 2’ and 7’ are integrals of the 
system of differential equations, 


da’ dy’ 


—_——~ = —— = (a) da, 
E(w’, y’)  n(@’, y’) 


(98) 
which for a = a, take on respectively the values z and y. Conversely, whatever 
the functions £ (a, y), n(x, y) may be, the equations x = f(a, y, a), y = o (a, y, a), 
which represent the integrals of the preceding system which reduce respectively 
to x and to y for a particular value a, of the parameter, define a continuous group 
of transformations. In the first place, it will be simpler to introduce a new 
parameter, 


t= if a (a) da, 


which enables us to write the differential equations (93) in the abridged form 
deen dy 


94 eer (eh aaa REE Be 
( ) E(x’, y’) n(x’, y’) 


a 
s 


II, § 35] INFINITESIMAL TRANSFORMATIONS 89 


The general integral of this system can be written, as we have seen above (§ 31), 
in the form 

Q,(v’, y’) = CO, Q, (v’, y’) =t + C2, 
where Q, and Q, are definite functions of x’, y’, and where C,, C, are two arbi- 
trary constants. The solutions which take on the values 2 and y fort = 0 are 
given by the system of equations 


(95) Q, (x, Y)= 2, (x, Y), Q, (x’, y’)= Q, (x, y) +t. 


The preceding expressions, indeed, define a continuous group, for if we carry 
out in succession the two transformations which correspond to the values t, 
and t, of the parameter, the resulting transformation corresponds to the value 
t, +t, of the parameter. The two transformations which correspond to the 
values t and —¢ are the inverses of each other. If we have 


w=f(t,y;t), yY=o ty; 4, 
we may write also 
t=f(r,y;—t), yao, y; —d. 
If we take for the new variables 
u=2,(@, y), v= 04(2, ¥), 
the equations (95) become 
(96) U’ =U, v=rvt+t; 


and we say that the group is transformed to the reduced form. Every con- 
tinuous one-parameter group is therefore similar to a group of translations. 

Let us take, for example, the group x’ = az, y’ = a®y. Applying the general 
method, we have 


02,” 7 7 
Fg De gare oy 


oa a oa 


y’ 
— 2 — 9". 
ay iy 


The differential equations (93) are in this case 
ites Oo da 
okie Da)? eg, So 


dt, 


where t= loga. The finite equations of the group can be written in the form 


7 


aoe log a = loga +f, 


and they will be brought to the reduced form by taking for the new variables 
log x and y/z?. . 


35. Application to differential equations. Let us consider a given differential 
equation 


dy dy a) 
97 F(a, y,—:—>::> =0 
7) (9 dx dx? nace , 


and a known one-parameter group of transformations of the form (86). Let us 
suppose that the equation (97) is identical with the equation obtained by carry- 
ing out on it the change of its variables x and y defined by the relations (86), 
whatever may be the numerical value of the parameter a. If this is the case, 
we shall say that the differential equation (97) admits the group of transforma- 
tions (86). We can make use of this property to simplify the integration. In 


90 EXISTENCE THEOREMS [II, § 35 


fact, let us suppose that we carry out a change of variables such that the equa- 
tions which define the group in question are brought to the simple form w’ = u, 
v’=v-+a. The same change of variables, applied to the proposed differential 
equation, leads to a new differential equation of the nth order, 


dv d?v d™v 
(98) &(u, 0, oes Te) =O, 


which does not change if we replace in it v by v + a, whatever may be the 
numerical value of the constant a. This can happen only if the left-hand side 
@ does not contain the variable v. If the equation is of the first order, we obtain 
the general integral by a quadrature. If n >1, we can lower the order of the 
equation by unity by taking dv/du for the new unknown dependent variable. 
Let us consider, for example, the homogeneous equation of the first order, 


This equation does not change if we replace x and y by az and ay respectively, 
whatever may be the constant a. Now the formule a’ = az, y’ = ay define a 
group of transformations, which can also be written in the form 


—=-, logy=logy +t. 


Hence if we put y/x = u, log y = v, we are led to an equation that is integrable 
by a quadrature (see § 3). 

Let us now consider linear equations of the first order, and first of all the 
homogeneous equation dy/dz + Py=0. Since this equation does not change 
when we replace in it y by ay, whatever may be the constant a, we say that 
it admits the group of transformations 2 = a, y’ = ay. Hence it will be inte- 
grable by a quadrature if we take log y for the dependent variable. 

Next, let 

dy 

(99) ae + Py + ¢ c=) 
be the general linear equation of the first order, and let y, be a particular 
integral, not zero, of the equation dy/dz + Py=0. It is easily verified that 
the equation (99) does not change if we replace y by y + ay,. Hence it admits 
the group of transformations defined by the equations 


7 


Liaw, Lee + a. 
Ub tiki 

Taking for the new dependent variable y/y,, the equation must reduce to an 
equation integrable by a quadrature. We are led to precisely the calculations 
of § 4, and it is easy to see in a similar manner that the different cases of reduction 
of the order of the equation which have been indicated in § 19, for equations of 
higher order, are essentially only particular cases of the preceding method. 

These different methods, which appear at first sight as so many different 
devices for solution, having no relation one to another, can thus be considered 
from a common point of view by means of the theory of groups of transforma- 
tions. To every continuous one-parameter group of transformations on the 


II, § 36] INFINITESIMAL TRANSFORMATIONS 91 


two variables « and y we can make correspond in this way an infinite number 
of equations of the first order which can be integrated by a quadrature, and 
equations of higher order whose order can be depressed by unity. 

This fact may be of practical importance in the setting up of the equations 
in certain problems. Suppose that it is a question of finding the plane curves 
which possess a certain property, and that we know a priori a one-parameter 
group (G) of transformations such that, if we apply any transformation of (@) 
to a curve having the given property, the new curve also has the same prop- 
erty. It is clear that the differential equation of these curves will admit the 
given group of transformations. If, then, we choose a system of coérdinates 
(u, v) such that the equations of the group (G@) shall become u’ = u, v’ =v +a, 
the differential equation of the curves sought in this system of codrdinates will 
contain only u, dv/du, d?v/du2,..-. For example, suppose that we wish to obtain 
the projections on the zy-plane of the asymptotic lines or the lines of curvature 
of a helicoid, the axis Oz being the axis of helicoidal movement in the sliding 
of the surface upon itself. It is clear that if a curve C of the zy-plane is a 
solution of the problem, then all the curves which we obtain by making C turn 
through any angle about the origin are also solutions. The differential equation 
of these curves admits, then, the group formed by the rotations about the origin ; 
the equations of this group in polar codrdinates are p’ = p, vw =w+a. With 
the system of variables p, w, the differential equation will contain, therefore, 
only p and dw/dp (see I, § 243, 2d ed.; § 242, Ist ed.). 

So far we have supposed the group G known. We are now led to examine 
the following problem: A differential equation being given, to recognize whether 
or not it admits one or more one-parameter continuous groups of transformations, 
and to determine these groups. This is a very important question, which cannot 
be developed here in detail. We shall limit ourselves to a few particulars. 


36. Infinitesimal transformations. Given a system of transformations on n vari- 
ables, defined by the equations 


(100) x; = fi (x, Loy tty Xn; a), (j=1, 7 RSG (4 


where the functions f; depend upon an arbitrary parameter a, we say again that 
these transformations form a group if the transformation resulting from any 
two transformations of the system carried out in succession itself belongs to 
the system. As above, a group is said to contain the identical transformation 
if, for some value a, of the parameter, we have 


Si(®. Voy ests En 5 My) = <i, i=], Beans n), 


for all values of 2, %,+++, 2n. It may be shown, as above, that such a group is 
obtained by integrating a system of differential equations 


day dx; dx; 
(101) SS en " 
Ei(Zy, Loy +++, Xp) Ey (@y, LQ, +++, Ly) En (1, +++, Ln) 
Let 
(102) Py Oi has ey © * 5a ss (i= 1, 2). in) 


be the integrals of this system which reduce to 2,, %,--+, %, respectively 
for t=0. The relations (102) define a continuous one-parameter group, the 


92 EXISTENCE THEOREMS (I, § 36 


variable t playing the part of the parameter. Indeed, we have seen (§ 31) 
that the general integral of this system can be written in the form 


7 7, / 
Q) (21, a, +++, L,) = Cy, Page 
id 4 7 7 7 / 
Qn —1(Ly, Lyy +++, X,) = Cn—1, On (Lj, Lay+++, L,) = b+ Ch, 


where ©, Q,,-+-+, Q, are n functions of the variables z;, which we have defined 
exactly. The integrals which for t= 0 take on the values 2,, %,--+, Xp are 
furnished, therefore, by the equations 


ee aN Z,,) = 0; (2, meARD) Ln), (i = ule 2, PR Le 1), 


(103) } , 
2, sey L,,) = Qn (£4, soles Zn) + ty 


which are equivalent to the relations (102). In this new form we see immediately 
that these transformations form a group. 

Let F(a, %,+++, fn) be a function of the n variables 2;; if we replace the 
variables 2; in it by the functions 2; given by the relations (102), the result 
F(z}, %,°°°, £,) is # function of x, %.,+++, tn, t, which for t= 0 reduces to 
F (a1, 2.,+++,; Ln). Let us consider the problem of developing this function 
according to increasing powers of t. We shall denote by F”’ the result of 
replacing z; by x, in F(a,, @, +++, %n), and we shall put 

of of 
X(f) = (2, Lo, en one A Ae y En (©, Loy ***y eae 
where f is any function of 2, %,,-++, ,. Similarly, replacing the variables 2; 
by «;, we shall put 
/ 7 / / ta of” 
Xf’) = Ey (@y, Lay +++, Ly) 


/ 


On; 


7 i / of 
+ soot &, (2, pote aay 
Ox 


n 


With these preliminary definitions, we have, by the differential equations (101), 


dF’ ‘ Teo ; reg Ghee eee, 
Fe E11 ++ +s Fn) zee En (hq, °° +) Ln) 7S AP’). 
Ox; dx, 


n 


Likewise we have 


thal He Sia 
—— = —[X’ (F’)] = X’ [X’ (F’ 
ay = G(X (FY) = XX" (FYI, 
and in general 
dp Fr’ 
= X’(»)(F’ 
pr OP): 


where X’(»)(F’) denotes the result of the operation X’ carried out p times in 
succession. Since, for t= 0, x}, %,+++, @, reduce to @,, %,---, tp, it follows 
that (dP F'’/dt?),—9 is equal to X‘») (Ff), and the development of F’ is given by 
the formula 

F’(ay, +++, €,) = F(@,, +++, Gn) + tX (F) 


2 
(102) + XO) t+ XO H... 
24 p! 


If we assume that the function F is regular in the neighborhood of the values 
£1, Ly,++*, Ln, the series on the right is convergent as long as |t| is sufficiently 
small. We have, in particular, 


, t {2 1K 


II, § 36] INFINITESIMAL TRANSFORMATIONS 93 


Let us give to ¢ an infinitesimal value ét. Putting dz, = x; — z;, and neglect- 
ing infinitesimals of higher order than the first with respect to 6t, the preceding 
formule can be written in the form 


(106) 5x, = g ot, bx, = &, ot, 0 Gp bin = &, 6. 


We say that these relations define an infinitesimal transformation and that X (/), 
_ or S£&; (@f/éx;), is the symbol of this infinitesimal transformation. To every one- 
parameter group corresponds an infinitesimal transformation, and conversely. 
If we choose at pleasure n functions, £,, §,-++, &, Of 1, 2, +++, Xn, the result- 
ing expression AX (f) is the symbol of an infinitesimal transformation that 
defines a continuous group whose finite equations would be obtained by inte- 
grating the system of differential equations (101). 

The introduction of infinitesimal transformations has made it possible to 
apply the methods of the differential calculus to the theory of groups. Besides, 
in many questions concerning groups it is the infinitesimal transformation 
which is concerned, as we shall see from a few examples. 

Let us consider 2,, 2,,-++, 2, as the codrdinates of a point in space of 
n dimensions, and ¢ as an independent variable which denotes the time. If ¢ 
varies, the point with the codrdinates xj, 75,---, 2, describes in a space of 
n dimensions a curve, or trajectory, starting from the point (7, +++, &). The 
space of n dimensions, or at least a region of that space, is thus decomposed 
into an infinite number of one-dimensional manifolds, and each point of the 
given region belongs to a single one-dimensional manifold. We say that a 
function F(x,,---, Z,) is an invariant of the group considered if we have 


F(z, Set 2) aa F(x,, 725 In), 


whatever may be the value of ¢. It is easy to obtain all the invariants of a 
group. In fact, dividing the two sides of the equation (104) by ¢, and supposing 
F’ = F, we obtain the relation 


t tp—1 
X(F) + 5 XO(F) + het hie AP) CH) eaves == 
p! 


Since this equality must hold whatever t may be, it is necessary in particular 
that we have X(f)=0. We say, in this case, that the function F admits the 
infinitesimal transformation of the group. This condition is, moreover, suffi- 
cient ; for if we have X (fF) = 0, we also have X [X (F)] = 0,---, and therefore 
X (p)(F) = 0, whatever p may be. The only invariants of the one-parameter group 
are therefore the integrals of the equation X(f) = 0. 

Let us notice that if two groups have for infinitesimal transformations X (/) 
and ILX(f) respectively, where II (x,, 7, +++, %,) is any function whatever, 
these two groups have the same invariants, even though they are not identical. 
If we apply to the same point the transformations of the two groups, this point 
will indeed describe the same path, but with different velocities. Conversely, 
if two groups have the same invariants, the two infinitesimal transformations 
A(f) and Y(f) can differ only by a factor II (x,, 22,+++, %,) which depends 
only upon @,, £,-++, £n, for the two equations X(f) = 0, Y(f) = 0 must have 
the same integrals. 

We shall now introduce another important concept. Let 


(107) =f, Y; 4), Y= (t, Y; 4) 


94 EXISTENCE THEOREMS [II, § 36 


be a continuous group in two variables. If we apply a transformation of this 
group to all the points of a plane curve C, we obtain another plane curve C,. 
Let y’, y”,---, y™ be the successive derivatives of y with respect to x and 
Yis Yi5++*, y™ the successive derivatives of y, with respect to z,; we have seen 
(I, § 61, 2d ed.; § 35, Ist ed.) how to calculate these last successive derivatives 
in terms of a, y, y’,---, y™. These calculations lead to formule of the form 


or = yy (x, Y, Vie a), 
(108) iene = ¥s os RE bi 


ah 


mo) — hy (ee yy y Cpa ak) roy a). 


The relations (107) and (108) define also a group of transformations in n + 2 
variables, x, y, y’,-+:, y™, which is called the extended group of the first. We 
shall assume this fact, the proof of which presents no other difficulties than the 
writing of rather long expressions. We shall merely show how the infinitesimal 
transformation of the extended group can be obtained. Let 


é F 
E(x, 0) A + ne no 


be the infinitesimal transformation of the given group. We can write the 
equations of this group in the form 


t t 
t= 04280, u) + ni (t+ noe) bene, 
(109) f ; On 
=v + Fale, v) +5 (822+ net) dons 


and from them we derive 
dy + - ‘(2 dz + — = ay) +>. 


Tea lara a "+o 
Ox 


The coefficient of t on the right, after expansion in a single power series, is the 
only thing we need to know. It is obtained by a division and is equal to 


SERGE. 9 2D 

ox oy ox/ dx oy/ \dz 
The symbol of the infinitesimal transformation of the extended group is, there- 
fore, forn =1, 


E(0, WW + n(2, ni 4 [a (Sys > 


The method is a general one. If the coefficient of t in the development of y{”—)) 


is w(x, Y, Y, +++, yY@—-D), we have for y™ 


t 
dyo- Do dy@™-D + rats Gat 


dz, a : (Fae 08 a v) ag 
log + oe OU 


yO = 


5 


II, § 36] INFINITESIMAL TRANSFORMATIONS 95 


and the coefficient of t on the right-hand side is 


da ar) (+ +Sy/. 
dx ox «(Oy 
Hence we can calculate step by step, to any desired value of n, the infinitesimal 
transformations of the extended groups which are obtained from the given group. 
We say that a system of differential equations 
(110) oy UE en 
Xx, * ere Xn 
admits the one-parameter group of transformations G defined by the equations 
(100) if it changes into a system of the same form, 


dz, dz, dz; 
(111) PG a Ot 
As BE xs 
when we take for new variables 2}, 73,--+-, Z, instead of @,, 2,,-++, %n, and if 


this is true whatever the value of the parameter a may be. Here and below, 
the symbol X; denotes the same function of the variables x; that X; is of the 
variables z;. In order that this be true, it follows from the relation which has 
been established between the system (110) and the partial differential equation 


wt i of 


soa ae aig ace ee 
On 


(112) X(f) = 
that it is necessary and A ee that every transformation of the group G shall 
carry the equation 

14 7 7 , ? ?, of 
Xx (f’) Sy x, (<4, Toy*t*s £4) — 
t= Ox; 
over into a linear equation equivalent to X(f)=0 for every value of the 
parameter a. If f(z,, 2, +++, @n) is an integral of X(f)=0, f’(x}, £g, +++, Z,) 
is also an integral of X’(f’) = 0; hence, if we replace x}, ---, x; by their values 
given by the expressions (100), f’(x], ---, Z,) must also be an integral of X (f) = 
It follows that the necessary and sufficient condition that the system of differ- 
ential equations (110) admit the group of transformations G is that every trans- 
formation of that group shall change an Sa ee of the equation X (f/f) = 0 into 
an integral of the same equation. 


Let 
(113) T(f) = a ade & $22 


be the infinitesimal cera of the group G. Replacing the parameter a 
by the parameter ¢ defined above, we may write 


/ / 4 t t2 
S(&, Toys ts ©) =f («,, Dost ts Ln) + 1 Nee ee T[T(Ff)] yrs 
If f(x,,-++, £,) is any integral of the equation (112), the same thing must be 
true of f(x}, +--+, £,), and consequently of 


SF (55 ++ *y Ep) —F (25 ++ +5 En) 


or of 


T(f) + - TET (AY + += %, 


96 EXISTENCE THEOREMS [II, § 36 


whatever the value of t may be. In particular, T(/) must be an integral of the 
equation (112). This condition is sufficient. For, let f,, f,,--+,,—-1 be a system 
of n — 1 independent integrals. If T(f,), T(4,), +++; T(fn-1) are also integrals, 
the same thing must be true of 7(f/), where f is any other integral. For we 
have f= II (f,, /,,-°°+,fn—1), and therefore 


oll oll 
T(f) te of, PGs) Ie toh ot pel Det) « 
Since T(f) is an integral, the same thing is true of T[T(/)], and so on; the 
same is therefore true of f(x}, @3,--+, 2). 

Hence, in order that the system (110) admit the group G of transformations, 
it is necessary and sufficient that, if f is an integral of X(f) = 0, T(f) shall also 
be an integral. We say for brevity that the equation X(f)=0 admits the 
infinitesimal transformation T(/). 

Let us now take a differential equation of the first order, 


d. 
(114) hg ie 
Anh ow 
In order that the equation X(f) =A df/ox + Baf/oy = 90 admit the infini- 
tesimal transformation & @f/éx + n af/dy, it is necessary that we have’ 
AS + Be a0, Een am le), 
Ox oy OL } 
where w(x, y) denotes an integral (other than a constant) of X(f)=0, and 
where II(w) denotes an undetermined function of w. We derive from these 
relations 
Ow BII (w) dw  —«- ATI (w) 
orb SAy ee BE ey) An = UBE. 


whence, 
dw Bdz — Ady 
(ay Pewsey al 


It follows that 1/(BE — An) is an integrating factor for Bdr — Ady. Conversely, 
let d(x, y) be a function such that its total differential is 


Pio Bdz — Ady 
BE — An 
Then we have, simultaneously, 
) ) ) ) 
X@=A 24+ B%=0, Te) =et4+,2=1; 
Ox oy Ox oy 


hence T(¢) is also an integral of the equation X(¢)=0. We can state this 
result as follows: 

In order that the differential equation (114) admit the group of transforma- 
tions derived from the infinitesimal transformation &ef/dx + n Of/dy, it is necessary 
and sufficient that 1/(BE — An) shall be an integrating factor for Bdx — A dy. 

This new method requires only the knowledge of the infinitesimal transfor- 
mation of the group. As there exist an infinite number of integrating fac- 
tors, we see that every equation of the first order admits an infinite number 
of infinitesimal transformations. 


II, § 36] INFINITESIMAL TRANSFORMATIONS Sh. 


Let us return to the general case of the system (110). Let X(f)=0 be 
the corresponding linear equation and T(f) the symbol of an infinitesimal 
transformation. Let us consider the equation 


(115) Z(f) = X[T(f)]—TLX(f)] =0, 


where X[7(f)] represents the result of the operation X(_ ) applied to T(f), 
and where T[X(f)] has an analogous meaning; Z(/) is still a linear homo- 
geneous function in the derivatives of the first order @f/dzr;, and it does not 
contain any derivatives of the second order. To show this, it suffices to prove 
that the coefficients of a derivative of the second order are the same in 
X[T(f)] and T[X(f)]. Now the coefficient of é2f/er? is X;é;, and that of 
O7f/0x; Ox, IS Xiée + Xz~& in T[X(f)], and it is obvious that these coeffi- 
cients are the same in X[T(f)]. The equation Z(f)=0 is therefore an 
equation of the same type as the equation X(f/)=0, which can be written 
in a form exhibiting the coefficients explicitly : 


(116) Z(f)=[X&)— mx te FEE Q)— TK 4... <0, 


If now T(/) is an integral of the equation XY(/) = 0, whenever f is an inte- 
gral of the same equation, every integral of X(f) = 0 evidently satisfies the 
linear equation Z(f) = 0; we must have, therefore (§ 31), 

(117) X(T (S)]— TIX SY] = p 1, Fo +++, En) X(F), 
where p is an undetermined function of 7,, %,-++, Z,. Conversely, if we have 
an identity of this form, every integral of the equation X(/) = 0 satisfies also 
the equation X[7(/)] = 0, and therefore T(/) is also an integral of the equa- 
tion X(f)=0. The necessary and sufficient condition that the linear equation 
A (f) = 0 admit the infinitesimal transformation T(f) is expressed by the relation 
(117), where p is any function of £1, %.,+++,%n- That relation is equivalent to 
the (n — 1) distinct relations 


X(&)— T(Xy) _ X(&)— (Hs) _ , _ XE) — TX). 
x, AG. AG 
Given a differential equation of the nth order, 


dry dy d?y —) 
118 < if) aD ey ; 
fn dan o(z Y Ga? dat?” dan} 


in order to determine whether it admits the group of transformations deduced 
from the infinitesimal transformation é (x, y) @f/ox + n (x, y) éf/oy, we need only 
replace the equation (118) by a system of n differential equations of the first 
order, taking for the auxiliary dependent variables the first (n —1) deriva- 
tives y’, y”,-+-, y*-), and then determine whether this system admits the 
infinitesimal transformation of the extended group of G. 

Let us consider, for example, the equation of the second order y” = ¢ (2, y,y’), 
which may be replaced by the system 


dz _dy___ay 
: 1 yy o(t, 4, Y’) 
or by the linear equation 
ms os @ 


¥ Ue 
cr Pee —y’ x, p) ar 
A (f) aE ae the YY) 


98 EXISTENCE THEOREMS — [II, § 36 


It will be necessary to determine whether this equation admits the infinitesimal 
transformation 

E(x, yes n(Z, ne + [t+ (2-S)r- | 
On carrying through the calculations, we find a condition which contains a, y, 
and y’, and which must be verified for all values of these variables. The equa- 
tion of the second order being given, if we wish to find the infinitesimal trans- 
formations which it admits, we have at our disposal the unknown functions 
E(x, y), n(x, y), Which do not contain y’. Writing the condition that the preced- 
ing relation is independent of y’, we may have, according to the given function 
(x, y, y’), a limited or unlimited number of equations which must be satisfied 
by the functions &(z, y) and (a, y). In general these equations will be incom- 
patible, and we see that an equation of the second order, taken arbitrarily, 
does not admit any infinitesimal transformation. The same thing is true of 
equations of higher order, and it is seen from this how Sophus Lie was able to 
classify differential equations according to the number of independent infini- 
tesimal transformations which they admit. 


EXERCISES 


1*, Let M, be the greatest absolute value of f(x, y)) when x varies from 2 
to 2, + a. If the letters a, b, K, %), yp have the same meaning as in § 30, the 
integral of the equation y’ = f(x, y) which takes on the value y, for x = q, is 
continuous in the interval (x), £) + p), where p is the smaller of the two numbers 
a and log(1+ Kb/M,)/K. 

[E. LinpELor, Journal de Mathématiques, 1894. ] 


[The inequalities 


Ita = tana] < Mya—1 C= 20" 


are established step by step, as in § 27, and y, will remain between y, — b and 
Yo + 6, provided that we have eX@-™) <1+4 bK/M,.] 


2. Find two first integrals of the simultaneous systems of differential 
equations 


@) Hae @y-v@2=0, Zrymyte meso, 


Sp ne hye ORs 
(8) (z—y) rato (2— y)? aa ee 


(y) EE BE A Nel) fr ac ADP anata OL 
y@@ty) (@@—y)(2e+2y+z) xet+y) 


3. The expression 1/[y — af (y/x)] is an integrating factor for dy — f(y/z) dz. 


4, The general form of the differential equations of the first order which 
admit the infinitesimal transformation y@f/dx — xdaf/éay is 


ry —y 2 2 
= O(a + 4 
e+ yy’ ( ¥) 


Deduce from this an integrating factor. 


Il, Exs. ] EXERCISES 99 


5. Find the general form of the differential equations of the first order which 
admit the infinitesimal transformation 0f/éx + x (éf/dy); the infinitesimal trans- 
formation «a (@f/0x) + ay (af/dy). 

6. Find a group of transformations for the differential equation 
dy 
—=¢(t+a 
where a is constant, and deduce from it an integrating factor. 

7*, The differential equations of the elastic space curve, 

YZ’ jess zy” —_— 6x’ Aas 3 By, 
2g’ oes pe’ — by’ sas 4 Ba, 
ay a ae, — 62’ =. Qa, 
where a, B, 5 are constants, possess the two first integrals x’? + y’2 + 22 = 0, 


B(x? + y?) — 42’ =C’. We then obtain xz and y by the integration of a differ- 
ential equation of the second order. 


[Hermirte, Sur quelques applications des fonctions elliptiques (p. 93). ] 


CHAPTER III 


LINEAR DIFFERENTIAL EQUATIONS 
I. GENERAL PROPERTIES. FUNDAMENTAL SYSTEMS 


Linear differential equations have been studied more thoroughly 
than any other class. They possess a group of characteristic proper- 
ties which distinguish them sharply and at the same time simplify 
their study. Moreover, they appear in a great number of important 
applications of Analysis, and a preliminary study of them is very 
useful before undertaking the study of differential equations of the 
most general form. Except when otherwise expressly stated, we 
shall study here only those equations whose coefficients are analytic 
functions of the independent variable. 


37. Singular points of a linear differential equation. A linear differ- 
ential equation of the mth order is of the form 


d’y ay 
(1) Faw Huge aie tt was Cae ot e+ yy + O41 = 0, 
yy) Uy***) G4, being functions of the single variable x. Its inte- 
gration is equivalent to that of the system 
ad 1 — 
+ O,Yn-1 $0 + U1 Y, + OY + Ong = 9, 
2 . 
“ iy, ay eee io 6 eee 
dz Yy> dx sell ey 5) dx Yn-19 


obtained by taking for auxiliary dependent variables the first n —1 
derivatives of y. Let us suppose that the coefficients a, are analytic 
in a circle C, with the radius R and with its center at the point a, 
and let ¥, 5, Yoor+ +> ys’ » be a system of nm arbitrary constants. 
Applying to the equations (2) a general result established above 
(§ 23), we see that the equation (1) has an integral analytic in the 
circle C,, taking on the value y, for x=x,, while its first n—1 
derivatives take on respectively the values yo, Yo, +++, Ys’ » for «= x. 

We know also, from the general theory, that it is the only integral 
of the equation (1) satisfying these initial conditions; we shall say 
for brevity that it is defined by the initial conditions (a, y5 Yor Yoo 

100 


IL, § 37] GENERAL PROPERTIES 101 


ys—»). Let us now suppose at first, for definiteness, that the 
coefficients a; are single-valued functions of z, having in the whole 
plane only isolated singular points. Let Z be a path joining two 
non-singular points a, and X, and not passing through any singular 
point; the integral which is defined by the initial conditions (@,, y,, 
Yo ++, yS'—») 1s represented by a power series P(x — x,) convergent 
in the ele C, with the center x, and passing through the nearest 
singular point to x,. We can follow, by means of this series, the 
variation of the integral along the path Z as long as the path does 
not go out of the circle C,. If the path Z leaves the circle C, at a 
point a, let us take a point 2, on the path within the circle C, and near 
enough to @ so that the circle C, with the center x, passing through 
the nearest singular point does not le entirely within the circle C,. 
From the series P(# — #,) and from those which we obtain by suc- 
cessive differentiations, we can derive the values of the integral and 
of its first n — 1 derivatives at the point 2, Let y,, yj,---, y~» be 
these values; the integral of the ee (1), which is defined by 
the initial conditions (@,, y,, Yj, °--, y"~”), 18 represented by a power 
series P,(a — x,) convergent in the Aes C,. The values of the two 
series P (a — a,) and P,(« — #,) are equal in the part common to the 
two circles C, and C,, since they each represent an integral of the 
equation (1) satisfying the same initial conditions. It follows that 
the series P,(x — x,) represents the analytic extension in the circle 
C, of the analytic function defined in the circle C, by the series 
P(« —«x,). If all of the portion of Z. included between a, and X 
does not lie in the circle C,, we shall take a new point x, on the path 
within C,, and so on as before. 

At the end of a finite number of operations we shall certainly arrive 
at a circle containing the point X. In fact, let S be the length of 
the path Z and § the lower limit of the distance of any point of Z 
to the singular points. The radii of the successive circles used are 
at least equal to 8, and we can choose the centers of these circles in 
such a way that the distance between two successive centers is greater 
than 6/2. After p operations the length of the broken line obtained 
by joining these successive centers will be equal to at least pé/2. 
If we have pé/2 > S, the length of the broken line will be greater 
than the length of Z. Hence, after at most (p —1) operations, we 
shall have arrived at a circle containing all of the portion of Z included 
between the center of that circle and the point X. 

Recapitulating, we see that we can continue the analytic exten- 
sion of the integral as long as the path described by the variable 


102 LINEAR DIFFERENTIAL EQUATIONS [ III, § 37 


does not pass through any of the singular points of the coefficients a;. 
We know, therefore, a priori, what are the only points which can be 
singular points for the integrals of a linear equation. It may, how- 
ever, happen that a point a is a singular point for some of the coeffi- 
cients a, without being a singular point for all the integrals. In the 
particular case where the coefficients are all polynomials or integral 
functions, all the integrals are analytic functions in the whole plane ; 
that is, they are integral functions and they may reduce to polynomials, 


The reasoning may be extended also to the case where the coefficients a; have 
any singularities whatever, it being possible for these functions to be multiple- 
valued. If we start from a point 2), where these coefficients are analytic, and if 
we cause the variable x to describe a path L, along the whole length of which 
we can continue the analytic extension of the coefficients a;, we can like- 
wise continue the analytic extension of the integrals along this path. The power 
series which represent the integrals are convergent in the same circles as the 
series which represent the coefficients. 

These results are entirely in accord with those which we have deduced from 
the method of successive approximations (§ 28). 


38. Fundamental systems. Let us consider a linear equation which 
is also homogeneous, that is, not containing a term independent of y, 


dl d™—14 
OPO Cees 


dy 
yn a, dy! aaa 


Ae ae 


$e tay + ay =0, 


where F(y) denotes no longer a function of the variable y but the 
result of an operation carried out on a function y of the variable z. 
From the definition of this symbol of operation it is clear that, if 
Y» Yor *y Yp are any p functions of az, and C,, C,,---, C, any con- 
stants, we have the relation 


FCI Y Ss CoYo + Nd 2c Cy Yp) = CLF (\) att CLF (Y,) ns +C,F(Y,)- 


If ¥1) Yoo °°) Yp are integrals of the equation (3), then C,y, + C,y, + 
---+C,y, is also an integral, whatever the numerical values of the 
constants C;may be. If we know n particular integrals y,, y,, +++, Y, 
of the equation, we can therefore deduce from them an integral 


(4) a Cy, = Fe C.Ye airs iy. at Ce 


in the expression of which appear x arbitrary constants C,, C,, - ++, C,. 
We cannot conclude from this that the expression (4) really rep- 
resents the general integral of the equation (3); we must first assure 
ourselves that we can dispose of the constants C,, C,,---, C, in such 
a way that, for a particular value a, of a, different from a singular 


point, y and its first n —1 derivatives take on any values given in 


III, § 38] GENERAL PROPERTIES 103 


advance. For the sake of brevity, let us indicate by (y?), the value 
which the pth derivative of the particular integral y,; takes on at the 
point a. Setting the values of the integral y, and of its first n —1 
derivatives at the point x,, equal to these arbitrary quantities, we 
obtain a system of m linear equations to determine the constants C,, 
C,,+++, C,- The determinant of the coefficients of these unknowns 
must be different from zero. We shall denote by A(y,, y,, +++; Yn) 
the determinant whose elements are the functions y,, ¥,,+-+, Yn 
and their derivatives up to those of the (n — 1)th order: 


Y, Y, yee al 
UAV.) Aer Go (ty 
yor aR sk eg ae a (n— 


If this determinant, which is an analytic function of x in the whole 
region in which the coefficients a, are analytic, is not identically 
zero, let us choose for x, a point where this determinant is not zero. 
We can then determine the constants C; so that y and its first n — 1 
derivatives take on any initial values whatever for x,. Every inte- 
gral of the equation (3) is therefore included in the formula (4). 
We say, for brevity, that this formula represents the general inte- 
gral of the equation (3). The integrals y,, y,, ---, y,, such that 
the determinant A(y,, y,,--+, ¥y,) 18 different from zero, form a 
fundamental system. 

If this determinant is identically zero, some of the integrals y,, y,, 

- , y, can be deduced from the others. We shall say, in general, 
that n functions y,, y,,---, Yy, of the variable x are not linearly 
independent if there exists between these n functions an identity of 
the form 


(6) Cry: “i Cry coy ts Cay, ss 0, 


where C,, C,,---, C,, are constants not all of which are zero. In order 
that n functions ¥,, Y,°**, Yn Shall not be linearly independent, it 
is necessary and sufficient that the determinant A(Y,, Y.)**+) Yn) be 
identically zero. 

The condition is first necessary. For from the relation (6) we 
obtain the nm —1 relations of the same form, 


(7) Cy a Coy? ame Coy? ia 0, (p = 1, 2, ig at RD oar 1) 


between the derivatives of the first order, of the second order, etc., 
of the functions y;. Since the coefficients C, are not all zero by 


104 LINEAR DIFFERENTIAL EQUATIONS (II, § 38 


hypothesis, the equations (6) and (7) cannot be consistent unless 
the determinant A(y,, ¥,,-+ +; Y,) 18 identically zero. 

Conversely, suppose that A = 0, and suppose first that all the 
first minors of A relative to the elements of the last row are not 
identically zero, for example, that the cofactor of y*~», 


Y Y, Ee etd 
of 4 4 
pee hl Y2 ahs fe 
=o (n—2 2 
Usa NB eae) ic het pan ‘ 


is different from zero. Let A be a region of the plane of the variable x 
where the functions y; are analytic and where this determinant 6 
does not vanish. Let us put 


Ye IG, AY Baya 
(8) Yn Te An ate uh “haere te ioe Yn- 1) 
Peon BS oe 7 
ae oeees Kas yr D4 Kye eye filo ae 2). 
These n — 1 equations determine K,, K,,---, K,_, as analytic func- 
tions of x in the region 4, since K; is the quotient of an analytic 
funetion divided by the minor 8 which is not zero in A. These func- 
tions K,,---, K,_, satisfy also the relation 


(QQ) yf P= Kyl? + Keyl ++ + Ky ye, 
since A(¥,, Ya***) Yn) 18 zero at every point of A. Differentiating 


once each of the equations (8), and taking account of these same 
relations and of the relation (9), we find 


ea whet Bo 1Yn—1 = 9, 
Mea ites 
Kiyo D4, hae eee 2) — 
and consequently Kj = K,=--.=K,_,=0. The functions K,,---, 
K,,_, ave therefore constants, and we have indeed a relation of the 
form (6) between the n functions y,, y,,-+-+, Y,, Where all the coeffi- 
cients are constants and the coefficient C, is different from zero. 
Since this relation has been established in the region 4A, it follows, 
by analytic extension, that it holds in any region in which the func- 
tions ¥,, Yo) °* +) Yn exist and are analytic. 
It will be noticed that the minor 6 is precisely equal to 


AY) Ya + Yaa) 
If this minor 6 is identically zero without A(y,, ¥,, +++; Ya—2) being 
also zero, a similar argument would show that the functions y,, y,, 


IL, § 38] GENERAL PROPERTIES 105 


+++, Y,—-1 Satisfy a relation of the form (6), where C,,= 0, C,,_, # 0. 
Continuing in this way, we shall therefore surely arrive at a relation 
of the form (6), in which some of the coefficients may be zero. If, 
therefore, we know x integrals of the equation (3) such that A(y,, ¥,, 

+, Y,) = 9, one at least of these integrals is a linear combination 
with constant coefficients of the other integrals. It may also happen 
that these m integrals reduce in reality to p independent integrals 
[p<n-—1}]. Inorder that this may be the case, it is necessary and 
sufficient that all the determinants analogous to A which can be 
formed with p +1 of these integrals shall be zero, one at least of 
the determinants formed with p integrals being different from zero. 

The same lemma enables us also to prove that the general integral 
of the equation (3) is represented by an expression of the form (4). 
For, let (Y,, Y¥5)+ ++; Yn) be a fundamental system of integrals, and y 
any other integral. From the (n +1) equations 


F(y) = 9, F(y,) = 9, oS) F(y,) = 0 


we derive, by elimination of the coefficients @,, a,, ---, @,, an equation 
of condition which is no other than 


(10) AY, Yr» Yor 2 89 Yn) = 0. 


We have, therefore, between these n + 1 integrals, a relation of the 
form 
Cy + Clyne a CrYn = 9, 


where C, C,, C,,--+-, C, are constants not all of which are zero. 
Finally, C, the coefficient of y, is certainly different from zero, since 
the integrals y,, y,,+--, Y, are linearly independent. 


Every linear equation of the nth order has an infinite number of funda- 
mental systems of integrals. In order to obtain such a system, we need only 
take n integrals such that the determinant formed from the initial values of 
these n integrals and their first n —1 derivatives for a non-singular point 


©) is not zero. If (y,, Yg,---, Yn) iS a first fundamental system, the n integrals 
Y,, Yo,+++, Yn, given by the equations 
Yi = Ci1y1 + CinYo t+ +++ + CinYny (¢= 1, 2,---, n) 


where the coefficients c;, are constants, form a fundamental system, provided 
that the determinant D formed by the n? coefficients cz, is different from zero. 
We have, in fact, by the rule for the multiplication of determinants, 


ACY, Me ro 6) Y 2) OTN Re Hac "3 Yn)- 


It follows from this relation that the quotient [dA(y,,---+, yn)/dx]/A is the 
same whatever may be the fundamental system. We shall verify this by cal- 
culating this quotient. For this purpose let us observe that the derivative of a 
function F(x) is equal to the coefficient of h in the development of F(x + h) in 


106 LINEAR DIFFERENTIAL EQUATIONS [III, § 38 


powers of h. If we give to x an increment h, and if we replace each element. 
of the determinant A by its development, retaining only the terms of the first 
degree in h, we obtain the determinant 


y, + hy} Yat hyg «+: Yn + hy, 
yi + hy yathyy ++ y, + hy, 
yO—D thy yS—YD 4 Ay 2-. yO—D 4 hy™ 


The coefficient of h is the sum of n determinants which are obtained by tak- 
ing the coefficients of h in any row and the terms independent of hf in the other 
rows; n—1 of these determinants, having two rows identical, are zero, and 
there remains 


Y Ye 228 Yn 
CNY) Vane a Vay Te ee ee 
dx ye—2) y@—%) 9... yo?) 

yO — yaa y™ 


This result is true, whatever the functions y,,---, y, may be; if these func- 
tions are integrals of the equation (3), we can replace y{” in the last row by 
— ay" — .--—adny,, and similarly for the others. There remains, after 
developing with respect to the elements of the last row and taking account 
of the determinants which have two rows identical, 


dA 


11 —=--—a,A. 
(11) eke 
The quotient which we wish to calculate is therefore equal to — a,, and we 


derive also from the preceding result the value of the determinant 


where A, denotes the value of A for x =z). This expression for A shows that 
this determinant is different from zero at every non-singular point, if it is not 
identically zero —a result which we could also have obtained from the preceding 


properties. 
It should be noticed that every linear equation of which a fundamental 
system of integrals is (¥,, Y,,++*, Yn) can be written in the form. (10) 


Aly, Y49 Yor ** Yn) = 0, 


the coefficients containing only the integrals y; and their derivatives. This 
shows that any n linearly independent functions y,, Y,,+++, Yn, can always be 
regarded as forming a fundamental system of integrals of a linear equation. 


39. The general linear equation. A non-homogeneous linear equation 
can be written in the form 


d arty 


nd d 
(12) Fy) = a + yo + a $y tay = f(x), 


where the term independent of y has been isolated on the right-hand 
side. We shall also consider the equation formed by replacing f(x) 


iI, § 39] GENERAL PROPERTIES 107 


by zero; the resulting equation, F'(y) = 0, is called the corresponding 
homogeneous equation. If we know a particular integral Y of the 
equation (12), the substitution y= Y+ 2 reduces the integration 
of that equation to that of the homogeneous equation F'(z)= 0 by 
the identity F(Y+%)= F(Y)+ F(z). The general integral of the 
non-homogeneous equation is therefore represented by the expression 


(13) Bae Mas CY; ne CoV, artis CrYnd 


where ¥,, ¥,,+**, Y¥, are a fundamental system of n particular inte- 
grals of the homogeneous equation, and where C,, C,,---, C, are 
nm arbitrary constants. It often happens in practice that we can 
easily obtain a particular integral of a lmear non-homogeneous equa- 
tion, and in this case we are led to the integration of the homogeneous 
equation. The search for a particular integral is facilitated by the 
following remark, which we need only state: If f(x) is the sum of 
p functions f,(x), f,(@), +--+, f,(@), such that we know how to find a 
particular integral of each of the equations 


F(y) =f,(@); FY)=f,@), satis Fy) =f,(@), 


the sum Y,+ Y,+---+¥, of these p particular integrals is an 
integral of the aaa ton F( TY 

In general, if we know the general integral of the homogeneous 
equation, we can always obtain by quadratures the general integral 
of the non-homogeneous equation (supposing, of course, that the 
left-hand side is the same for the two equations). 

The following process, due to Lagrange, is called the method 
of the variation of constants. Let (y,, ¥,,-+++, Y,) be a fundamental 
system of integrals of the equation F(y)=0. Imitating as much 
as possible the process employed for a linear equation of the first 
order, we shall seek to satisfy the equation (12) by taking for y an 
expression of the form 


(14) i fee CLY, = Coy, ab ae a7 CrYns 


where C,, C,,---, C, denote n functions of x We can evidently 
establish between these m functions n — 1 relations chosen at pleas- 
ure, provided that they are not inconsistent with the equation (12), 
If we put 

YyCi HY gla tees + ynCr = 9, 

(15) ae ate ie re Gar HS =H, 


Shia 
e- D0; pray POS F by OC, =, 


108 LINEAR DIFFERENTIAL EQUATIONS [IIl, § 39 


the successive derivatives of y up to the (n —1)th derivative have 


the values ; ’ i , 
Y — Cin ae CoY2 ie 2S ERO 
(16) y oR Wee ote ae “> + fe 


ee 5 
yo A yyy set — (I ie 1) + C i as) ae? Te BR SS ono 

The first of the relations (15) has been chosen in such a way that 

the first derivative y' has the same expression as if C,, C,,---, C, 

were constants, and similarly for those that follow. The derivative 

of the nth order has a less simple form: 


Y = CY + Cry +--+ Cy? 
“bay Cree Ye) Cartan iiaay en uaa 
Substituting the preceding values of y, y', y", .- +, y™ in the left-hand 
side of the equation (12), the coefficients of C,, C,,---, C, are re- 
spectively F(y,), F(y,), +++,“ (Yn), and we are led to the new relation 


(15 YPC + YPPCE A FYS-PCL =F), 


which, together with the relations (15), enables us to determine 
Ci,-°-, Cz. We can therefore find:C,, C,,---, C, by quadratures. 


n 


We can also make use of the following method, due to Cauchy. 

Let (¥,, Yos***, Yn) be a fundamental system of integrals of the equation 
F(y)=90. Let us determine the constants C,, C,,---, C, so that the integral 
Cy, + +++ + CrYm and its first n — 2 derivatives all vanish, while the (n — 1)th 
derivative reduces to unity for a value @ of. The integral ¢ (x, a) thus obtained 
depends, of course, upon the variable x and also upon the initial value a, and 
satisfies the n conditions 


(17) (a, a) =9, (a, a) = 9, gp” (a, y= Oe Bes o"-) (a, a) =1, 
where $‘”) (a, a) denotes the pth derivative of ¢(x, a) with respect to x for the 


value «=a. If we replace a by z in the preceding relations, which amounts 
to a simple change of notation, they can also be written in the form 


(17) $(%,2)=9, ¢’ (x, x) = 0, see, G™—2%)(x, x) = 0, g@-) (a, «) =1, 


where ¢P) (xz, 2) denotes now the pth derivative of ¢(x, a) with respect to a, in 
which we have replaced a@ by @ after the differentiation. With this under- 
standing let us consider the function represented by the definite integral 


(18) vi if “$ (2, a) f(a) da 


with an arbitrarily fixed lower limit z). Applying the general formula for 
differentiation, and taking account of the conditions (17’), we find successively 


= foe, a) F(a) da, wee, —= fie p-1) (x, a) f(a) da, 
Mr = [79 (@, a)f(a)da + F(2. 


III, § 40] GENERAL PROPERTIES 109 


Substituting these values of Y, Y’,---, Yin F(Y), we find 
F(Y)=f(«) + i [p™ (x, a) + A, P%-D(X, &) + +++ + Ong (x, a)] f(a)da. 
X 


The function under the integral sign on the right is identically zero, since ¢ (a, @) 
is an integral of corresponding homogeneous equation, whatever may be the 
value of the parameter a. From this it follows that the function Y represented 
by the definite integral (18) is a particular integral of the non-homogeneous 
linear equation. It will be noticed that this integral, as well as its first (n — 1) 
derivatives, is zero for the lower limit 2), which is supposed different from a 
singular point.* 

The application of this method to the equation d"y/dx" =f (x) leads to 
precisely the result obtained above (§ 18). 


40. Depression of the order of a linear equation. If we know a certain 
number of particular integrals of a linear equation, we can make use 
of them to diminish the order of the equation. Let us consider first a 
homogeneous equation of the nth order, and let y,, y,, +--+, ¥,, (p< 7) 
be linearly independent integrals of that equation. The substitution 
y =¥,%, where z indicates the new dependent variable, reduces the 
proposed equation F(v)= 0 to a new equation of the same type in 
z, for the expression for any one of the derivatives d?y/dz? is itself 
linear with respect to z and to its derivatives. If y, is an integral 
of the equation F(y)= 0, the new equation in z must have z=1 
for a solution, which requires that the coefficient of 2 shall be zero; 
this fact is verified at once by calculation, for the coefficient of z is 
precisely F(y,). The equation in z is therefore of the form 

d"z Qe 2 dz 


1 dat—} 


0, 


*It is easy to verify that the method of the variation of constants and Cauchy’s 
method lead to the same calculations. In fact, the function ¢(x, @) of Cauchy is of 


the form 
p (x, &) = Py (BX) yy (&) + Po (A) Yo (x) + +++ + Gn (Q) yn (2), 


where the functions $;(@) are determined by the conditions 
Py (&) yy (@) + +++ + Pn (&) yn (@) =0, 
(A) Py (@) Y{~? (AX) + +++ + Gn (BY (a) =0, 
Py (&) YO —Y (B) to + Gn (A) YY (a) =1, 


and the particular integral (18) has the value 
x x 
Y= 112) f %(@F(@aat--+yn(z) f Fn(@M/(@) da. 
Sad) 0 


But if we compare the conditions (A) with the relations (15) and (15’) which deter- 
mine the Cc in the method of the variation of constants, we see at once that we have 
C; (x)= $i (x) f (x), and therefore the first method gives us a particular integral by 
the same quadratures. 


110 LINEAR DIFFERENTIAL EQUATIONS [III, § 40 


where 0,, 0,,--++, 0, _; are functions of a. This equation reduces to 
a linear equation of order n — I, 
ing ot am 4 
(20) i da*— + aly perma  gobuss 9 toy Waals 


by putting uw = dz/dx. If y,, y,,---, y, are p integrals of the equa- 
tion from which we started, the equation (19) has the p — 1 integrals 
Yo/Yr9***s Yp/Yy) and therefore the equation (20) has the integrals 


a Oa) ee eas 
da\y,]? a daXy, 


These p —1 integrals are lnearly independent; otherwise there 
would exist a relation of the form 


a [Ye 4 (Yp\ _ 
Cae (yt To de (7 ae 


where C,, C,,---, C, are constants not all of which are zero, and we 
could conclude from it, by integration, the existence of a relation of. 
the same form, C,y, ++-+-+C,Y, + C,y, = 9, where C, is a new con- 
stant. If p > 1, the application of the same process leads from the 
equation (20) to a new linear equation of order n — 2, and so on. 
The integration of a linear homogeneous equation of which p inde- 
pendent particular integrals are known reduces, therefore, to the inte- 
gration of a linear homogeneous equation of order n — p, followed by 
qguadratures. When p=n—1, the last equation will also be inte- 
grable by a quadrature. 

Similarly, if we know p integrals, y,, y,,--+, Y,, of a non-homoge- 
neous equation, such that the » — 1 functions 


Yo— Yr ie Yn — Y, 
are linearly independent, the substitution y = y, + 2 leads to a homo- 
geneous equation having the p —1 integrals y,— y,,---, % —Yy,- 


It is therefore possible to reduce this equation to a linear homogene- 
ous equation of order n — p +1. 
Consider, for example, the linear id of the second order, 


(21) FY) = ah pot pagers =O 
and let y, be a particular integral of this equation. If we put y = yz, 
we find 


dy dz dy, gy one Oe dy, dz d*y, 
Sema te dace daha oa ana © dee? 


Ill, § 40] GENERAL PROPERTIES ip by 


and, substituting in the equation (21), we find, since the coefficient 
of z is zero, 


(22) ye (2c +n) = 0. 


1 dx? dx dx 


Putting dz/dx = u, this equation can be written in the form 


whence, by integration, 
Logu+ f pds + Log yj = Log C, 
or : 
~ [pda 
pr anus 


A second quadrature will give z and consequently y. We see that 
the equation (21) has the integral y, given by the expression 


"de -f pda 
(23) i nf SigtCak we rs 


U 


vi 
which is independent of y,. The general integral of a linear homo- 
geneous equation of the second order is therefore obtained by two quad- 
ratures when we know a particular integral.* 

This property is a mark of similarity between the linear equation 
of the second order and Riccati’s equation (§ 7). There exists, in 
fact, a very close relation between these two kinds of differential 
equations. If we depress the order of the homogeneous equation (21) 
by the process of § 19, by substituting 


— oft 


“ss ? 
we are led to a Riccati equation, 
(24) z+2+pze+q=0. 


* We can derive from these results a very simple proof of an important theorem 
of Sturm. Let us suppose that the coefficients p and q are continuous real functions 
of the real variable x in the interval (a, b), and let x, 7; be two consecutive zeros of 
a particular integral ¥;(x) in the interval (a, 6). If ye(x) is another particular inte- 
gral independent of (x), the formula which gives u can be written 


d [a Crs pda: 
a us) Ress 
dx\y1/ yj 

which shows that the quotient 72/y1 varies always in the same sense when x increases 
from x, to x;. Now this quotient is infinite for c= 2% and for x=%,; hence it constantly 


increases from —« to + or decreases from +x to—«. The equation y2(x) =0 has 
therefore one and only one root in the interval (xo, x4). 


112 LINEAR DIFFERENTIAL EQUATIONS [III, § 40 


Conversely, any Riccati equation 
(25) u'taw+t+bu+tc=0, 
where a, 6, ¢ are functions of x(a # 0), may be reduced to the 


form (24) by putting w = z/a, which transforms (25) to an equation 
of the form (24), 


! 
a ee +-ac = 0. 
It follows that the general integral of the equation (25) is 
LCs Vie Yo 
OY ated 


where y, and y, are two independent A wes of the linear equation 


(26) Hee 


! 


y+ (6 = “)y + acy = 0. 


This expression really contains only a single arbitrary constant, that 
is, the quotient C,/C,, which appears in it to the first degree.* 


Example. Legendre’s polynomial X,, (I, § 90, 2d ed.; § 88, Ist ed.) satisfies 
the differential equation 


a 


(27) (l— U*) aaa — 228 + nin + 1yy =O. 
a: 


*It would seem that a quadrature might be necessary to derive the general 
integral of the linear equation (21) from the general integral of Riccati’s equation 
(24). In reality this is not the case, or, rather, the quadrature reduces to the calcula- 
tion of jk pdx. In general, let z= (x, C) be the general integral of a differential 
equation of the first order, dz/dx= SF (a, 2). From the relation 


as se (x, ?) 


we derive, by differentiating with respect to the constant C, 


io af dd |. af_2(,,, 26), 
eee do ac 5a (Les 5a 


From the last equation we find afi (6f/0p) dx = Log (0f/6C), where, of course, the 
same value of the constant C is to be understood in the two sides of the equation. 
Applying this to Riccati’s equation (24), if z=¢(«, C) is the general integral of that 
equation, we conclude that 


Op 
2 zde+ fi dz + Log (—+)=0. 
if p (<4) 


If 21, 2, Z3 are three integrals of the equation (24), on carrying out the calculation 
(see §7) we find that the general integral of the linear equation (21) has the form 


ot fed C1 (23 — 21) + Co (23 — Ze) 


Y — e 
V (21 — 22) (% — %3) (2g — 2%) 


III, § 41] GENERAL PROPERTIES 1138 


To prove this it suffices to notice that, by putting wu = (7?— 1)”, we have the 
relation (7? —1)u’ = 2nzu, and by taking the (n+ 1)th derivative of the two 
sides we have an equation which is identical with the equation (27) when we 
replace d"u/dz" in it by y. In order to obtain a second particular integral of 
the equation (27), we shall apply the general formula (28) with p equal to 
22/(x? —1)=1/(e + 1) + 1/(e— 1); this gives 


dx 
tone Xa ———————_—_————— 
ty = Xn f (c? — 1) X2 


It might seem necessary to know the n roots, 1, %q,++*, An, Of the polynomial 
AX, in order to calculate this integral, but this is not the case. For, let us write 
the integrand in a form which exhibits the simple fractions which come from 
the roots + 1 and — 1 of the denominator : 


1 Dy ek ( Lp ee ) Py 
(@@—1)x2 2\e—-1 41) x2 
where P,, is a polynomial of degree 2n — 2, the quotient obtained by dividing 
1— X? by x? —1. It follows that 
*) LX: AGe " dz. 


This last integral is a rational function, for if it contained a logarithmic term 
such as Log(x — aj), the point a; would be a singular point for y,, and the inte- 
grals of the equation (27) can have no other singular points than « = + 1 (§ 37). 
We can therefore calculate this integral by rational operations (I, § 104, 2d ed. ; 
§ 109, Ist ed.). Since the integral must be of the form Q,-1/Xn, where Qn—1 
may be taken as a polynomial of degree not greater than n — 1, we can deter- 
mine the coefficients of this polynomial, for example, by the condition 


ieee. ae Qn—1+n = Bae 


1 
Yo — 58 Log = 


Having once obtained the polynomial Q, 1, we may write the general integral 
of the equation (27) in the form 


—] 
tae C, Xn + C5] Qa +5 + —X,, Log (= a 


41, Analogies with algebraic equations. The preceding properties establish an 
evident analogy between the theory of linear differential equations and the 
theory of algebraic equations. This analogy persists in a large number of 
questions. As an example of this we shall show how we can extend to linear 
equations the theory of the greatest common divisor. In general, let 


d Yy dv—ly 
10 Gan ga dgn—1 
be a symbolic polynomial where a), @,,---+, @, are given functions of xz. If a, 
is not zero, we shall say for brevity that F(y) is of the nth order. If G(y)isa 
symbolic polynomial of the same nature and of the pth order, it is clear that 
G[F(y)] is again a symbolic polynomial of the same kind and of the (n + p)th 
order. Let now 


dry 
Fy) = pes + 6, 


F(y)= 


dy 
Steal a len Y 
dx 


qn-1 d 
ae foes + Om 1 + OmY 


114 LINEAR DIFFERENTIAL EQUATIONS [III, § 41 


be another polynomial of order m(m=n). We can find a third polynomial G (y) 
of order n — m such that F(y) — G[F,(y)] is at most of order m — 1 (a poly- 
nomial of order zero is of the form ay, where a is a function of x). Let us put 


qd —my qr—m—ly 
OL hi Te ara net awh Oe me 


Gy) =a 
The coefficient of d”y/dx" in G[F,(y)] is \9b), and if we take \, = a)/b,, the 
difference F(y) — \,d"—™[F,(y)]/da—™ will be at most of order n—1. Let aj 
be the coefficient of d"-1y/dx"—1 in this difference. If we take \, = aj/bo, the 
difference 


Fy) — <= [FW] — s 5 LA) 


Ors ore 7 ate 
will be at most of order n — 2. Continuing in this way, we see that we can 
determine, step by step, the coefficients \,, \,,-++, 4,—m in such a way as to 
obtain an identity of the form 


(28) Fy) — GF, (y)1= F,(y), 
where Ff, (y) is at most of order m — 1. This operation is entirely analogous to 
the division of one algebraic polynomial by another. 

Now suppose that we wish to obtain the integrals common to two linear 
equations 


(29) F(y)=0, Fy) =0 


The identity (28) shows that these integrals are the same as the integrals 
common to the two equations f(y) = 0, F,(y) = 0. If F,(y) is not identically 
zero, the same operations can be repeated on F\(y) and F,(y), and so on 
until we arrive at two consecutive polynomials, f,_1(y) and F;(y), such that 
Fy -1(y) = Ge_-1[Fe(y)]. This last symbolic polynomial F;(y) is the analogue 
of the algebraic greatest common divisor: all the integrals common to the two 
equations (29) satisfy the linear equation Fy (y) = 0, and conversely. If Fy (y) is 
of the degree zero, the two equations have no other common integral than the 
trivial solution y = 0. | 

If in the relation (28) F,(y) is identically zero, the equation F(y)= 0 has 
all the integrals of F, (y) = 0. Conversely, in order that F(y) = 0 shall have all 
the integrals of F,(y) = 0, it is necessary that F,(y) be identically zero, for a 
linear equation of order not greater than m— 1 cannot have all the integrals 
of a linear equation of the mth order. Hence in this case we have identically 


Fy) = G[F\Y)], 


and if we put F,(y) = z, the integration of F'(y) = 0 is reduced to the successive 
integration of the two linear equations 


G(z)=0, FLY) =2, 


of orders n— m and m, of which only the second is non-homogeneous. 

We can deduce from this observation another solution of a problem already 
treated. Suppose that we know p independent integrals y,, y., +++, Yp(p <n) 
of F(y)=0. We can form a linear equation of the pth order having these p 
integrals (§ 38). Let F,(y) = 0 be this equation of the pth order; then we have 
identically F(y) = G[F,(y)], and if the equation G(z) = 0 of order n — p has 


III, § 42] GENERAL PROPERTIES 115 


been integrated, we can integrate F,(y) =z by quadratures alone, since we 
know the general integral of F,(y)=0. The reduction is the same as by the 
first method, but the new process is more symmetric. 

Appel, Laguerre, Halphen, E. Picard, and many others after them have ex- 
tended to linear equations the theory of symmetric functions of the roots, the 
theory of invariants, and the very fundamental work of Galois relative to 
the group of an algebraic equation. The theory of invariants is founded on the 
easily verified fact that a linear homogeneous equation is changed into a new 
equation of the same kind by every transformation of the type 


c=f(t), y=2z¢(t), 


where ¢ is the new independent variable and z the new dependent variable, 
whatever the functions f(t) and ¢(t) may be. 

We can sometimes make use of this transformation to simplify a linear equa- 
tion. For example, if we wish to make the coefficient of the derivative of 
order n—1 disappear, we find that it suffices to put 


ye sen J 18 


b) 


retaining the variable z. Since we have two arbitrary functions f and ¢ at our 
disposal, it would seem that we could take advantage of them to make two 
coefficients disappear; but this reduction, although theoretically possible, is 
illusory in most cases. For example, we can always choose the functions f and 
¢ so as to reduce any linear equation of the second order to the simple form 
z’’ = 0, but the actual determination of these functions presents the same diffi- 
culties as the problem of integrating the original equation. 


42, The adjoint equation. Lagrange extended the theory of integrating factors 
to linear equations in the following way. Let F(y) be a linear function of y 
and of its first n derivatives, 


F(y) = agy™ + ayy@-D 4 +++ An-1Y + On; 


where a, @,,+++, @ are any functions of z, and where y’, y”,---, y™ denote 
the successive derivatives of y. Let us try to find a function z of x such that 
the product zF(y) shall be the derivative with respect to x of another function 
linear in y and in its derivatives up to those of order n—1. The general for- 
mula for integration by parts (I, § 87, 2d ed.; § 85, Ist ed.), applied to each of 
the terms of the product zF(y), gives us 


atc: d dr—-1(a, | 
ak Wipe (4-1) red (n—2) 4. vss 2 
EU) Em Yes | ave dg (10%) ¥ ae nL Netra 
d is d bat dn —2(a, 2) 
(30) + S| ae 2) — ae (a,z)yC-9) +--+ Fy Paka 
+ Cem emer err eeere reser ee eres ereseserseseresseeneses 
d 
+ Fy IPD + yG(z), 
where we have put 
n qn-1 d (An - 
(31) G(z)=(— rye oe) 4 = 1-12 2 a vo net) 4 Ay Z 


116 LINEAR DIFFERENTIAL EQUATIONS [III, § 42 


If we denote by W (y, z) the expression which appears on the right-hand side 
of the equation (80) which is bilinear with respect to y and z and to their 
derivatives, we can write that equation in the abridged form 


(32) Fy) —uG@) = 1%, 9) 


so that for all the possible forms of the functions y and z the binomial 
zF (y) — yG@(z) is the derivative of W(y, z). If we now take for z an integral of 
the equation G(z) = 0, the product zF'(y) is the derivative of an expression of 
the same form, linear with respect to y, y’,---, y*—-), and the equation 
F(y) = 0 is equivalent to a linear equation of order n— 1, 


(33) V (y, z)= G, 


which we obtain by replacing in ¥ the function z by the integral in question. 
Now the equation G(z) = 0 is likewise a linear equation of the nth order; it 
is called the adjoint equation of F(y) = 0, and the symbolic polynomial G (z) is 
called the adjoint polynomial of F(y). 

We see, then, that if we know an integral of the adjoint equation, the inte- 
gration of the given equation is reduced to the integration of a linear equation 
of order n — 1 whose right-hand side is an arbitrary constant. If we know p 
independent integrals, z,, 2,-+++, Zp, of the adjoint equation, every integral of 
the given equation satisfies p relations of the form 


(34) W(y,z,)=C,,  V(y,%,)=Cy, +++, VY, %)= Op, 


where C,, C,,--+-, Cp denote p constants. Eliminating the derivatives y*—, 
ym—2),..., ye-p+) from these p equations, we obtain a linear equation of 
order n — p whose right-hand side depends upon the p arbitrary constants C,, 
C,,--+, Cp. In particular, if p = n (that is, if we know the general integral of 
the adjoint equation), we can solve the n equations (34) for y, y’,---, y*—, and 
we can obtain the general integral of the given eauation without any quadrature. 

There exist between the integrals of the two equations F(y) = 0, G(z)=0 
some remarkable relations, which we cannot develop here.* We shall only show 
that there exists a reciprocal relation between these two equations. More pre- 
cisely, if G(z) is the adjoint polynomial of F(y), then, conversely, F(y) is the 
adjoint polynomial of G(z). Forif F,(y) denotes the adjoint polynomial of G(z), 
we have a relation between F,(y) and G(z) of the same form as the relation (32), 


(32) yG (2) — 2F\v) =~ [%,(,2)]. 
Adding the relations (82) and (82’), we find 
FW) — FW = IY, 9+ Mas DI. 


If F(y) — F,(y) were not zero, the product z[F(y) — F,(y)] would be the deriva- 
tive of a function containing z and some of its derivatives. Now the derivative 
of a function containing z, 2’,---, z@) contains at least one derivative of z, 
namely, z?+, The preceding relation is therefore possible only if F,(y) is 
identical with f(y). 


* See DARBOUX, Théorie des surfaces, Vol. Il, Bk. IV, chap. v. See also Exercise 
17, p. 171, at the end of this chapter. 


11, § 43] SOME PARTICULAR EQUATIONS 18 bis 


II. THE STUDY OF SOME PARTICULAR EQUATIONS 


43. Equations with constant coefficients. Linear differential equa- 
tions with constant coefficients were integrated by Euler. Consider 
first a homogeneous equation 


(85) FY) = YM + ay—P+---+a,-1y' + Hy = 9, 


where @,, @,,+-+-,@, are any constants. By the general theory (§ 37) 
none of the integrals of this equation have a singular point in the 
finite plane; that is, they are integral functions of x. Let 


2 


x x Bay 
(36) Yer Cr Og dy he decate Omi ha a 


be the development in series of an integral. The series which repre- 
sent the successive derivatives have an analogous form. Replacing y 
and its successive derivatives by their developments in series in the 
left-hand side of the equation (35), and equating to zero the coefficient 
of any power of a, say x”, we obtain the following relation between 
n +1 consecutive coefficients : 


(37) Ca+p as A,on+p—-1 + Aon +p —2 ‘am hehe = An—1%p41 ax Anln a 0. 


If we substitute in it successively p = 0, 1, 2,---, we can calculate, 
step by step, all the coefficients ¢,,c, 4 ,-+-+, 1m terms of the n first 
coefficients ¢,, ¢,, +--+, ¢,-1, which may be taken arbitrarily. The 
series (36) thus obtained is convergent in the whole plane and repre- 
sents the integral which for 2 = 0 is equal to c,, while the first n —1 
derivatives take on respectively the values ¢,, ¢,,---, ¢,_, for x = 0. 
We shall show that this integral can be expressed in terms of expo- 
nential functions when it does not reduce to a polynomial. 

The equation (37) is a recurrent formula with constant coefficients 
which connects the n + 1 consecutive coefficients. Now it is easy to 
find particular solutions of that equation. For this purpose, let us 
consider the algebraic equation 


(38) fMart+ar—i+ar-?+.---+a,17 +a, = 9, 


which, for the sake of brevity, we shall call the auxiliary equation, 
the left-hand side f(7) being the auxiliary polynomial. If r is a root 
of this equation, it is clear that the relation (37) is satisfied, what- 
ever may be the value of the integer p, by putting ¢,,=7”". The 
particular integral thus obtained is equal to e, and we see that e” 
is a particular integral of the equation (35) if r is a root of the 
auxiliary equation f(r) = 0. The verification is immediate, for if we 


118 LINEAR DIFFERENTIAL EQUATIONS [ III, § 43 


replace y by e” in the left-hand side of the equation (35), the result 
of the substitution is e f(7). 

If the equation (38) has n distinct roots 7,, 7,,---,7,, we know n 
particular integrals e”, e”:",---, e’n*, and therefore an integral 


(39) y — C; ene + Ce" + ans 4+ Coen, 


the expression for which contains n arbitrary constants C,, C,, +++, Cy. 


This expression represents the general integral, for the determinant 
A (e"", es", +--+, e’n*) can be written in the form 


2 n—1 
iL Tr; ry a Cit ry 
a r re see ea: 
A= eitret ss tm& 2 ; : 5 
2 n—1 
1 oe TOES fo are 


and the determinant on the right is, except for sign, the product of 
the differences 7; — 7. 

Before studying the case in which the auxiliary equation has 
multiple roots, we shall prove a lemma. Let us make the substi- 
tution y = e**z in the equation (35), where @ is any constant and 2 
the new dependent variable; by Leibnitz’s formula we have 

y! = e** (az + 2), 
SP penn OM ice 


(40) y? — coors + * aPp-1z! + pea “P-2ztl +- eee + 2”), 


Substituting these valués of y, y', y",--- in the left-hand side of 
the equation (35), e** appears as a factor, and we have 
Fé" 2) =e7G (2), 

where G(z) is a linear expression in 2, 2',---, 2 with constant 
coefficients. In order to calculate the coefficients of G(z), let us 
observe that if we replace in F(y) the indices which indicate differ- 
entiation by exponents, and y itself by y° =1, the result obtained is 
identical with f(y). If we carry out the same transformation with 
the function z, the formule (40) may be written symbolically 


y? = 6" (@ + 2); 
hence G(z) can also be written, in the same symbolic notation, 


f(«+ 2), and, replacing the exponents of z by the indices which 
indicate differentiation, we see that the new equation in z is 


(41) F(em2) =e| M@et f@e't FO a4... + Ow] <0 


III, § 43] SOME PARTICULAR EQUATIONS 119 


Now let 7 be a p-fold root of the auxiliary equation; if we replace a 
by that root 7 in the equation (41), the coefficients of z, 2’, 2". . -, 2@-) 
in this equation are zero, and we obtain an integral by taking for z a 
function whose pth derivative is zero, that is, an arbitrary polynomial 
of degree p —1. Consequently, ifr is a p-fold root of the auxiliary 
equation, to that root corresponds p particular independent integrals of 
the linear equation (35), e", xe", +++, aP—te™. 

Let the & distinct roots of f(r)= 0 be 7,, 7,, +++, 7, and let their 
respective orders of multiplicity be denoted by p,, M,, +++, M.(3¢;= 7). 
From these roots we can form n particular integrals of the linear 
equation. These n integrals are independent, for any linear relation 
with constant coefficients between these n integrals would lead to an 
identity of the form 


8h, (0) + eG, (2) + + HG. (@)= 0, 


where ¢$,, $,,°-°, ¢, denote polynomials not all of which vanish 
identically. Such a relation is impossible if the k numbers 7, 7,, - - -, 
r, are distinct. For, let n,, ,,---, m be the respective degrees of 


these polynomials. It is understood that any term in the identity 
is simply omitted if the corresponding polynomial is zero. Dividing 
by e”, we can again write this relation in the form 


p, (@) + ea", (x) + cee t er", (4) = 0. 


Differentiating both sides of this equation, we have 


;(“) a ea)" hy (x)+ (7, oe r) , (x) ] +---=0. 
The degree of the polynomial which multiples e~"?* is again 
equal to n,, and the polynomial does not vanish; and similarly for 
the others. After having differentiated (m, + 1) times, we shall have, 
therefore, a relation of the same form as the relation from which 
we started, but with one term less, 
era (a) + ee? (@) + + + ey, @) = O, 

where the A —1 numbers s,,---, s, are different, and where y,, y,, 

- +, y, are polynomials of degrees n,, n,,---, 2, respectively. Continu- 
ing in this way, we arrive finally at a relation of the form er (x) =0, 
where 7 (x) is a polynomial not identically zero. But this is evidently 
absurd. The general integral of the linear equation (35) is therefore 
represented by the expression 


(42) ae Ee tesa g tsi acre, Ay 19 


“where P, 1, *++, Py,-1.are polynomials with arbitrary coefficients, 
of degrees equal to their subscripts. 


120 LINEAR DIFFERENTIAL EQUATIONS [III, § 43 


If the auxiliary equation has imaginary roots, the general integral 
(42) contains imaginary symbols, but we can make these imaginaries 
disappear if the coefficients a,, a,,--+-, a, are real. For in this case, 
if the equation f(7) = 0 has the root a + fi of multiplicity p, a — Bi 
is also a root of the same degree of multiplicity. The sum of the two 
terms of the formula (42) coming from these two roots can be written 


e**[ (cos Ba + isin Bx) & (x) + (cos Bx — isin Be) ¥(x)], 


where ®(x) and W(x) are two polynomials of degree p—1 with 
arbitrary coefficients, or in the equivalent form 


e**[ cos Bx®,(x) + sin Bx, (x) ], 


where ®, and ¥, are also two arbitrary polynomials of degree p — 1. 


Note. In order to express the general integral of the equation (85) in terms 
of exponential functions, we observe that it is first necessary to solve the equa- 
tion f(r) = 0. If this equation is not solved, the recurrent relations (87) enable 
us always to calculate, step by step, as many as we wish of the coefficients of the 
power series which represents the integral corresponding to the given initial 
conditions. 

We can determine in advance the number of coefficients which it suffices to 
calculate in order to obtain the value of the integral with a certain degree of 
approximation. Let A be the largest of the numbers 1, |a,|,-+-, |a@n|, and B the 
largest of the numbers |C |, |¢,|,-++, |¢n—1]. It is easy to prove, step by step, 
that we have |¢,+,|<B(An)P?+1,. The absolute value of the remainder of the 
series which represents the integral, commencing with the term in ¢” +, is there- 
fore less than the value of the series 


nie (An)P+2pn+p+1 vf, 
(n + p)! (n+ p+1)! 
where p =|z], and consequently less than 
B(An)p +1 px tp 


Anp , 
(n+p)! F 


Consider now a non-homogeneous linear equation with constant 
coefficients. We can avoid the use of the general method and find 
a particular integral directly if the right-hand side, $(«), is a poly- 
nomial. For if the coefficient a, of y in the equation 

d”"y ay 


dy ae 
Jan Ft Gann bit tun Zt ayy = bya” + b,a"— 2 oe 


is not zero, we can find another polynomial of degree m, 

y= (a) =eu"+ ca™-*+..., 
which, substituted for y in the left-hand side of the preceding 
equation, gives a result identical with @(a). The m +1 coefficients 


III, § 43] SOME PARTICULAR EQUATIONS 121 


Cy) Cyy Cyy ***y Cm are determined, step by step, by the relations 
anc, = 5, AnC, + MA, 16, = 4,, 

Ane, + (m —1)a,_1¢, + m(m —1)a,_.6, = 4, rey 
where a,, is different from zero by hypothesis. This computation is 
not applicable when a, = 0. More generally, suppose that the deriva- 
tive of the lowest order which appears in the left-hand side is the 
derivative of the pth order. Taking for the dependent variable 
z = d’y/dx”, the given equation is transformed into a linear equation 
of order n — p, where the coefficient of z is not zero. According to 
the case which has just been treated, this equation in z has a poly- 
nomial of the mth degree for a particular integral. Hence one par- 
ticular integral of the equation in y itself isa polynomial of degree 
m+ p. The coefficients of this polynomial can again be determined 
by a direct substitution. It should be noticed that the coefficients of 
x?—*, ~?-?, ..-, x, and the constant term are arbitrary. 

If the right-hand side ¢(«) is of the form e**P (a), where @ is con- 
stant and P(a#) denotes a polynomial, we reduce this case to the 
preceding by putting y = e**z, which leads to the equation 
eee Par: 

1A Ok (n —1)! dx 
This equation has for a particular integral, .as we have just seen, a 
polynomial whose degree we can determine a priori; the equation 
in y has therefore a particular integral of the form e*”Q(a), where 
Q(x) also is a polynomial. Suppose in particular that P(x) reduces 
to a constant factor C. If @ is not a root of the auxiliary equation, 
the equation (43) has the particular integral z = C/f(a), and the equa- 
tion in y has the particular integral Ce**/f(a). If a is a multiple 
root of multiplicity p of the auxiliary equation, the equation (43) is 
satisfied by putting 


$e $f) E+ f@2 =P) 


Pp 
f(a) FS =p C, 
| Seco? 
f(a)’ 
and consequently the equation in y has the particular integral 
Ca? e*/f (a). By virtue of a general remark (§ 38) we can there- 
fore find a particular integral directly whenever the right-hand side 
is the sum of products of exponentials and polynomials. This is the 
case in particular if the right-hand side is of the form P(2) cos ax 
or P(x) sin ax, for we need only express cos aa and sin ax in terms 
of e* and of e~. Having once recognized by the preceding 


or 


rod 


122 LINEAR DIFFERENTIAL EQUATIONS [ III, § 43 


considerations the form of a particular integral, it is not necessary 
to pass through all the indicated transformations in order to calculate 
the coefficients upon which it depends; it is often preferable to 
substitute directly in the left-hand side of the given equation. 


Example. Let it be required to find the general integral of the equation 


4 
(44) Py) =—4 — y = aer + be + csing +9 cos 2a, 


where a, b, c, g are constants. The auxiliary equation r#—1=0 has the sim- 
ple roots 1, —1, + i, —i; the general integral of the homogeneous equation is 
therefore 

(45) y = C,e + Coe-* + C, cosz + C, sing. 
We must next find a particular integral of each of the four equations obtained 
by taking successively for right-hand sides ae*, be?*, c sin x, g cos2a. Since unity 
is a simple root of f(r) = rt — 1 = 0, the first of these equations has the particular 
integral axe*/f’(1) = axzet/4. Since 2 is not a root of the equation f(r) = 0, the 
second equation has the particular integral be?”/f(2) = be?”/15. 

In the third equation, F(y) =c¢ sina, we can replace sin g by (e* — e—*)/2%, 
and we have to seek a particular integral of each of the two equations 


FW) = 556", Fy) =- 5 e-%. 

Now, since + i and —i are simple roots of f(r) = 0, we know, a priori, that they 
have respectively two particular integrals of the form Mae™, Nxe-**. The sum 
of these two integrals is of the form 2(m cosz + nsinz), and we can determine 
the coefficients m and n by substituting in F(y) and equating the result identi- 
cally to csing. This method avoids the use of the symbol 7. It turns out that it 
is necessary to take m =c/4, n= 0. We find similarly that the last equation 
F(y) = g cos2a has the particular integral g cos22/15. Adding all these par- 
ticular integrals to the right-hand side of the equation (45), we obtain the general 
integral of the given equation (44). 


44. D’Alembert’s method. A large number of methods have been 
devised for the integration of linear equations with constant coeffi- 
cients, particularly in the case where the auxiliary equation has mul- 
tiple roots. One of the most interesting, which is applicable to many 
questions of the same kind, consists in considering a linear equation, 
in which f(r) = 0 has multiple roots, as the limit of a linear equation 
in which all the roots of f(r) = 0 are distinct. In general, let 

dy d™-ly 


a dy x 
(46) F(y)= di” +, a, dx” —} aes elt Dy Atiiny — 0 


be a linear equation, where the coefficients a,, a,, ---, a, are functions 
of « which depend also upon certain variable parameters @,, @,- ++, @. 
Suppose that there exists a function f(a, 7) having the following prop- 
erty: for g values of r, depending upon the parameters @,, @,, +--+, @, 


III, § 45] SOME PARTICULAR EQUATIONS 123 


and in general distinct, the function f(a, r) of x is an integral of the 
equation (46). Let 7,, 7,,---+,7, be these ¢ values of 7 such that the 


sina I (@, 1); I (&; ".); sina 53 I (&; rm) 


form g independent particular Sei of the equation (46), whatever 
the values of the parameters @,, a,,---, @, may be. If for certain par- 
ticular values of these ai ing q er alues Tims) ity are Dob 
distinct, the number of the known integrals is diminished. Suppose, 
for example, that r, becomes equal to r,. If 7, is different from r,, 
the equation has the two integrals f(x, 7,), f(a, 7,), and consequently 
I (@, 12) —F(&; 71) 
ee 

is also an integral. Now, if 7, approaches r,, the preceding function 
has for its limit the derivative [ f(z, 7)],. If a third root 7, becomes 
equal to r,, we take, similarly, supposing first that 7, differs a little 
from r,, the integral 


S(@; Ts) ACE AAC ay CREE? r) I,, 
(75 7G ”)” 
and this integral has for its hmit [ f(z, 7r)],,/2 when 7, approaches r,,. 
This reasoning is perfectly general: if, for certain values of the par- 
ameters @,,---, @,, k of the roots are equal to r,, the corresponding 
equation (46) has the # particular integrals 


tors (E.G) > G4), 


In the case of a linear equation with constant coefficients the 
parameters @,, @, ---, a, are the coefficients themselves, and the 
function f(x, r) is e”. This leads again to the results which we 
obtained before directly. 


45. Euler’s linear equation. The linear equation 


(47) ot Aa oe +A, ae Ay = 0, 


where A,, A,, ---, A, are constants, reduces to the preceding by the 
change of variable* x = e’. Since dt/dx =1/x, we have 


dy _ dy dt 1dy d*y =(<4 <1) 


dx dt dx «x dt ae? oe Nea = dt 


* The general theory (§ 37) tells us that the integrals of the equation (47) can have 
no other singular point than x=0. Now e? cannot be zero for any value of ¢. The 
integrals obtained by the change of variable x = e‘ must therefore be integral functions. 


124 LINEAR DIFFERENTIAL EQUATIONS [IIl, § 45 


and we easily verify, step by step, that the product x?[d?y/dx”] is 
a linear expression with constant coefficients in dy/dt, d’y/dt’, 
..+, d?y/dt?. The given linear equation is therefore transformed by 
this change of variable into an equation with constant coefficients. 

To obtain the general integral of the equation (47), it is not 
necessary to carry out the calculations of this change of variable, 
for we know that the transformed equation has integrals of the 
form e’. The given equation has therefore a certain number of 
integrals of the form (e’)”= 2”. Replacing y by «” in the left-hand 
side of the equation (47), the result of the substitution is x’f(r), 
where 

fm=r@—1)--9@—n+1) 
+A, r(r—1)---@—n+2)+---+4,_17 +4,. 


If the equation /(r)= 0, which here plays the same role as the 
auxiliary equation, has n distinct roots r,, 7,,---, 7,, the general 
integral is | 
7 = Cia" pe er tl Pld oe IE oe 


Ifr is a multiple root of multiplicity w of f(r)=0, to that root 
corresponds, by D’Alembert’s method, the mw particular integrals 


ot —1yr 
ort —1 


0 
ie ap (305) "ORs wie gas = a” (Log a)*—1. 


The general integral of the equation (47) is therefore in all cases 
(48) y= 2 P, _,(Logx)+---+ 2 Py,-1 (Log), 


where 1, 7,, +++, 7, are the k distinct roots of f(r) = 0, where pu, “,, 
-+, #, are their orders of multiplicity, and where P,,_, (Log x) is 
a polynomial in Log x with arbitrary coefficients of degree mu; — 1. 
If, in the equation (47), we replace the right-hand side by an 
expression of the form «"Q(Log x), where Q denotes a polynomial, 
it can be shown, as in the case of the equations with constant coeff- 
cients, that the new equation thus obtained has as a particular inte- 
gral an expression of the same form, whose unknown coefficients 
can be calculated by a substitution. 


46. Laplace’s equation. We can sometimes represent the integrals of a linear 
equation by definite integrals in which the independent variable appears as a 
parameter under the integral sign. One of the most important applications of 
this method is due to Laplace and affects the equation 


dn—ly 


me +++ + (dn + Onx)y = 0, 


(49) FW) = (ay + 02) 4 + (ay + 2,2) 


III, § 46] SOME PARTICULAR EQUATIONS 125 


whose coefficients are at most of the first degree. Let us try to find a solution 
of this equation by taking for y an expression of the form 


(50) y= it Zex dz, 
(L) 


where Z is a function of the variable z and where L is a definite path of inte- 
gration independent of z. We have, in general, 


ask a ZzP eX dz, 

daxp (L) 
and, replacing y and its derivatives in the left-hand side of the equation (49) by 
the preceding expressions, we find 


51 | F(y)= { Zex(P dz 
(51) (y) =f Zer(P + Qe) de, 
where we have set, for brevity, 


P= Az" + a, 2*-14 +++ + An_-12 + Gn, 
Q = 092" +b, 2%-1 + +++ 4+ 0,12 + On. 


The function under the integral sign in the expression (51) is the derivative 
with respect to z of Ze?” Q, provided that we have 


BR 


d(ZQ) _ d cr 
(52) T= 2P, or ZF [Log (ZQ)] = ai 


dz 


We derive from this condition 


where the lower limit z, does not cause Q(z) to vanish. The function Z having 
thus been determined, the definite integral (51) is equal to the variation of the 
auxiliary function 
zP 
V=eZQ= endl eo" 

along the path Z. It will suffice, therefore, in order to obtain an integral of 
the given equation (49), to choose the path of integration Z in such a way that 
the function V takes on the same value at the end as at the beginning, and so 
that the integral (50) has a finite value different from zero. 

Let a, b, c,--+, 1 be the roots of the equation Q(x) = 0. The auxiliary func- 
tion V is of the form 


(53) V = et +R (z — a)*(z — D)B.-- - (2 — IA, 


where F(z) is a rational function whose denominator has no other roots than 
the roots a, b, c,---, 1 of Q(x), and of a multiplicity one unit less. Let 4, B, 
C, --- denote loops described about a, b, c,---, in the positive sense, starting 
from an arbitrary initial point, and let 4_;, B_1, C_1,--- denote the same loops 
described in the opposite sense. The function V is multiplied by e?7** when z 
describes the loop A, and by e—27'¢ when z describes the loop _1, and simi- 
larly for the others. It follows that if we make the variable z describe the 
loops A, B, A_1, B_, in succession, the function V takes on again its initial 
value. The definite integral (50), taken over this path ABA_1B-1, is not, in 


126 LINEAR DIFFERENTIAL EQUATIONS [III, § 46 


general, zero. It gives a particular integral of the given equation. By associat- 
ing the p points a, b, c,-+-, J in pairs in all possible ways, we obtain p(p —1)/2 
integrals, which in reality reduce to p — 1 independent integrals. 

We do not find n particular integrals in this way. In order to obtain others, 
we may look for the paths L having their extremities at certain of the singular 
points a, b,c, +--+, and such that the function V vanishes at the two extremities. 
If a is a simple root of Q(x) = 0, the function Z contains the factor (z — a)*—}, 
and it will be possible for the integral (50) to have a finite value when one of the 
extremities of the path ZL is at the point a only if the real part of @ is positive, 
and in this case V does approach zero at the same time as |z— a]. If a is an 
m-fold root of Q(z) = 0, the rational function R(z) contains a term of the form 
Am—1/(z — a)™—1. In order to determine the behavior of the absolute value of 
V in the neighborhood of the point z = a, we need only study the absolute value 


of the following important factor : vas 
m— 


amy a ohz—aym—1 
Setting (a Na : 


z—a=p(cos@?+ising), Am-1=A(cosy+isiny), a=a+a%, 
we may write the absolute value of this factor in the form 
e— ah pe eAp!—” cos[y —(m— 1) 6], 


In order that V shall approach zero with |z — a], it will suffice to make z de- 
scribe a curve such that the angle @ which the tangent makes with the real 
axis satisfies the condition cos [Wy — (m —1)¢] <0. For example, we may take 
@=[¥+(2k+4+1)7]/(m—1). If the angle ¢ has been taken in this way, the 
product Ze?” also approaches zero with |z—a]|. Proceeding in the same way 
with the other points b, c,---, 1, we see that we can determine new paths L, 
closed or not, giving other particular integrals. 

Finally, we can also take, for the paths of integration, curves going off to in- 
finity. We are again led to determine a path LZ having an infinite branch such 
that the function V approaches zero when the point z goes off indefinitely on 
this branch. If, for example, the rational function R (z) is zero, and if the angle 
of z lies between 0 and 7/2, it will suffice to make z describe an infinite branch 
asymptotic to a line that makes an angle of 37/4 with the real axis. 

Leaving these general considerations,* let us consider in particular Bessel’s 
equation, 


dy dy 


where n is a given constant. We have here 


P-=(2n +1)z, Q='1 + 2%, 
and consequently 
1 


1 
Z=(1+2) 2, V=e(14+22)""2. 
The definite integral 


Hae 
- (55) y= ip endl + 2)" Fae 


* See an important paper by Poincaré in the American Journal of Mathematics, 
Vol. VII. 


III, § 46] SOME PARTICULAR EQUATIONS 127 


is therefore a particular integral of the equation (54) if the function 


1 

err (1 + 22)" "2 
takes on the same values at the extremities of the path of integration. We can 
first take a succession of two loops described, the first in the positive sense 
about the point z =+ 7, the second in the reverse sense about the point z =— i. 
For the second path of integration we can take next a curve surrounding one 
of the singular points + i and having two infinite branches with an asymptotic 
direction such that the real part of zz approaches — o. 

The real part of the constant n may be supposed positive or zero, for if we 
put y = z—2"z, the equation in z does not differ from the equation (54) except 
in the change from n to —n. When this is the case, we can also take for the 
path of integration the straight line joining the two points + 7 and — 7%. More- 
over, the integral thus obtained is identical with the first except for a constant 
factor. In order to reduce this integral to the usual form, let us put z= it. It 
then takes the form 


+1. a a 
y= fo e(l— ey" 2a, 
or 


+1 wa 
(56) yf cos at (1— #2)" 2dt. 
-—1 


The remarkable particular case in which n is half an odd number deserves 
mention. If n is positive, the integral (56) always exists and can even be cal- 
culated explicitly, since n — 1/2 is a positive integer. But if the path L is a 
closed curve, the definite integral (55) is always zero. It seems, then, that in this 
case the application of the general method gives only one particular integral. 
However, in this apparently unfavorable case we can express the general inte- 
gral in terms of elementary functions. For, let us make the inverse transfor- 
mation to the preceding, so that n shall be half of a negative odd number. Then 
n — 1/2 is a negative integer, and the definite integral (55), taken along any 
closed curve, is a particular integral of the linear equation (54). Taking for the 
path L a circle having one of the points + i for center, we see that the residue 


of the function t 


ext (1 4 zy" 2 


with respect to each of these poles is an integral of the linear equation. Now, 
it is clear that the residue with respect to the pole z =+ i is the product of e” 
and a polynomial, and, similarly, that the residue with respect to the pole z =— i 
is the product of e-** and a polynomial. These two particular integrals are 
independent, for their quotient is equal to the product of e?** and a rational 
function. It is clear that their sum is a real integral, as is also the product of 
their difference and i. 


Note. The linear equation with constant coefficients is a particular case of 
Laplace’s equation, which is obtained by supposing all the coefficients b; zero. 
If we suppose also a, = 1, we have Q(z) = 0, while P(z) reduces to the auxil- 
iary polynomial f(z). The general method appears to fail, since the expression 
for Z becomes illusory. But it requires only a little care to recognize how the 
method must be modified. In fact, the reasoning proves that the definite integral 


128 LINEAR DIFFERENTIAL EQUATIONS [III, § 46 


f L Ze*« dx is a particular integral of the linear equation, provided that the defi- 
nite integral (7, Zf(z) edz, taken along the same path L, is zero. Now, if we 
take for L a closed curve, it is sufficient that the product Zf(z) be an analytic 
function of z in the interior of this curve. If, therefore, II(z) denotes any 
analytic function in a region R of the plane, the definite integral 


\¥ II (z) d 
c irs cae 


taken along any closed curve L lying in this region, is a particular integral of 
the linear equation with constant coefficients. We see how this result, due to 
Cauchy, is thus easily brought into close relation with Laplace’s method. 

As a verification, it is easy to find the known particular integrals. Let z=a 
be a p-fold root of the auxiliary equation f(z) = 0. Let us take for the path of 
integration a circle about a as center not containing any other roots of f(z) = 0, 
and let II (z) be an analytic function in this circle. The residue of the function 
II (z) e@*/f (z) or II (z) e**/[(z — a)? Ff, (z)] is equal to the coefficient of hp —1 in the 
development of the product 


ah 
II (a + h) eax PES es 
F(a + h) 
according to powers of h. If we have 
(a +h) _ 
f(a + h) 
the coefficients A,, A,, +--+, Ap—1 are arbitrary, since the function II(z) is any 
function analytic in the neighborhood of the point a. The residue sought is 
therefore equal to 
| gp-i oe oe 
Ere eS ee Jose 
Co in aoe 
that is, to the product of the exponential e** and an arbitrary polynomial of 
degree p—1. This agrees with the result already known. 


A,+A,h+ soe Ap_yhp-14+ tes, 


SF ot Aya], 


III. REGULAR INTEGRALS. EQUATIONS WITH 
PERIODIC COEFFICIENTS 


Aside from the very elementary cases which we have just treated, it is, in 
general, impossible to determine, simply from the form of a linear equation, 
whether the general integral is algebraic or whether it can be expressed in terms 
of the classic transcendentals. Mathematicians have therefore been led to study 
the properties of these integrals directly from the equation itself, instead of 
trying to express them (somewhat at random) as combinations of a finite num- 
ber of known functions. We have already seen (Chap. III, Part I) that the 
nature of the singular points of an analytic function is an essential element 
enabling us in certain cases to characterize these functions completely. We 
know a priori (§ 37) the singular points of the integrals of a linear equation. 
We shall now show how we can make a complete study of the integrals in the 
neighborhood of a singular point in a special case, which is nevertheless rather 
general and very important. 


IL, § 47] REGULAR INTEGRALS 129 


47. Permutation of the integrals around a critical point. Let a be an isolated 
singular point of some of the coefficients p,, p., +++, Pn of the linear equation 


dry dn—ly 
(57) UAL spec iysanmea 


7 ere, my 2 0. 
dx 

We shall suppose also that the coefficients are single-valued in the neighborhood 
of a. Let C be a circle with the center a in the interior of which p,, p., +++, Dn 
have no other singular point than a and are otherwise analytic. Let x be a 
point within C near a. All the integrals are analytic in the neighborhood of the 
point £). Let y,, Yo,°++, Yn be n particular integrals of a fundamental system. 
If the variable x describes in the positive sense a circle passing through the 
point z) about a as center, we can follow the analytic extension of the integrals 
Vis Yor °**s Yn along the whole of this path, and we return to the point 2) with 
n functions Y,, Y,,---, Y, which are again integrals of the equation (57), 
where Y; indicates the function into which y; passes after a circuit around 
the point a in the positive sense. We have, therefore, since Y,, Y.,---, Yn 
are integrals of the equation (57), n relations of the form 


ae = Oy Vy + Uy Yq Free + Minn, 
(58) Von = Gq Y1 + Gen Yo + °°° + AanYny 
BAA IS aS GCN ial ah aR a aah 


Yn = Gn1Yit In2yot -+++ Ann Yny 


where the coefficients a; are constants which of course depend upon the fun- 
damental. system chosen. It is easy to obtain the value of the determinant D 
formed by these n? coefficients. For we have, by § 38, 


x 
- d. 
A(¥;; Youi-*s Unj = Ce ere a 


If z describes the circle y with the center a@ in the positive sense, y; changes 
into Y;; hence we have 


sip yorche 
A(Y,, Yaak: <5 Yn) = A(Y4s Yoo °° +5. Yn) € Sy? : 


But the quotient of the two Wronskians is equal to D (§ 38), so that D = e— 272, 
where R indicates the residue of p, with respect to the point a. This determi- 
nant is therefore never zero. 

Since the coefficients in the equations (58) depend upon the fundamental 
system chosen, it is natural to seek a particular system of integrals such that 
these expressions are as simple as possible. Let us seek first to determine a 
particular integral u =), Yy, + A Yo + +++ + AnYn, Such that a circuit around the 
point a reproduces that integral multiplied by a constant factor. It is necessary 
for this that we have U = su, where U is the value of u after the circuit, and 
where s is a constant factor, that is, 


Vy (44494 xy A40U% pees 6 AinYn) + See 
ae Xn(An1 Y1 S On2Y2 Salis Onn Yn) — S(\1Y1 ees An Yn) == 0; 


Such a relation cannot exist between the n integrals unless the coefficients 
Of Y1, Yo, ‘++, Yn all vanish separately. The n+ 1 unknown coefficients 


130 LINEAR DIFFERENTIAL EQUATIONS [ III, § 47 


Ay» Agy ++) An, S Must satisfy the n conditions 


Ay (441 — 8) + AQ My +e + Andi = 0, 
(59) Ay Ae + Ag (Agq — 8) + +++ + Andina = 0, 
A1 in +A2den © +:++++An(Qnn— 8) = 0. 


Since the quantities \,, \,,+++, An cannot all be zero at the same time with- 
out having u = 0, we see that s must be a root of the equation of the nth degree, 


ay1i— § a21 oss Ant 
a2 deg — § BEG An2 
(60) F(s) = “|= 0, 
“Ain don ita aa Ann — 8 


which we shall call the characteristic equation; according to a remark made a 
moment ago, this equation cannot have the root s = 0, for the determinant D 
of the n? coefficients a; would be zero. 

Conversely, let s be a root of this equation; the relations (59) determine 
values for the coefficients \; not all zero, and the integral u =, y, + +--+ Ann 
is multiplied by s after the circuit around the point a. This being the case, let 
us suppose first that the characteristic equation has n distinct roots 8,, 8), +++, 8p. 
We shall have n particular integrals u,, u,, +++, U, such that, after the circuit in 
the direct sense around the point a, we have 


(61) U;, = 3%, Le camisetas see, Uy = Stn, 


where U; denotes the final value of u; after the circuit. These n integrals u,, Ug, 
-+, Un form a fundamental system. For, suppose that we have a relation of 
the form ; 


(62) Cu, + Cou, + -+- + Cru, = 9, 


where the constant coefficients C,, C,,---, C, are not all zero. After one, two, 
+++, (n—1) circuits, we should have the relations of the same form, 


wee + pas + ese Cn Sn Un man. 
(63) O,87U, +08, tees + OnSitn = 0, 
C1 Ra tty + Cy Boe Op St oly 0 
The linear relations (62) and (68) can be satisfied only if we have at the same 
time C,u, = 0,--++, Cru, = 0, since the corresponding determinant is different 
from zero, 

It is easy to form an analytic function which is multiplied by a constant 
factor s different from zero after a circuit around the point a. In fact, the func- 
tion (« — a)" or e” Log @—2) jg multiplied by e2™* after such a circuit, and if we 
determine ‘r by the condition r = Log (s)/2 zi, this function (x — a)” is indeed 
multiplied by s after a circuit around a. Every other function u having the 
same property is of the form (x — a)"¢(x — a), where the function ¢ (zr — a) is 
single-valued in the neighborhood of the point a, since the product u(% — a)-” 
comes back to its initial value after a circuit around the point a. The integral 
uz is therefore of the form 

Uz = (& — A)"* Gy (XZ — Q), 


IIL, § 48] REGULAR INTEGRALS 131 


where 7; = Log (sz)/2 wi and where the functions ¢, are single-valued in the 
neighborhood of the point a. In a circle C with the radius R about the point a 
as center and in which the coefficients p,, ---, Dy, are analytic except at the 
point a, the integral uz, cannot have any other singular point than a. The same 
thing is therefore true of the function ¢, (x — a), and the point a is an ordinary 
point or an isolated singular point for that function. We can dismiss the possi- 
bility that a@ is a pole. In fact, if the point a were a pole of order m, since the 
exponent 7; is determined except for an integer, we can write 


Uy = (t= ayre-™ (aw — a) ox — a], 


and the product (© — a)" ¢,(x — a) is analytic for «=a. If the point a is not 
an essentially singular point for ¢;(2 — a), we say that the integral is regular for 
«=a. We can then suppose that the function ¢,(z — a) has a finite value, 
different from zero, for r= a. 


48, Examination of the general case. It remains to examine the case where the 
characteristic equation has multiple roots. We shall show that we can always 
find n integrals forming a fundamental system and breaking up into a certain 
number of groups such that if ¥,, ¥.,+++, Yp denote the p integrals of the same 
group, we have, after a circuit in the positive sense around the point a, 


(64) Y,=sy,, Vy =8(Y1 + Yo); es Yp = 8(Yp—1 + Yp)- 


The different values of s are the roots of the characteristic equation, and to the 
same root may correspond several different groups. If the n roots are distinct, 
which is the case we have just examined, each group is composed of a single 
integral, 

The problem reduces in reality to showing that we can reduce the linear sub- 
stitutions defined by the equations (58) to a canonical form such as we have 
just indicated by replacing y,, yo, +++, Yn by suitably chosen linear combinations 
of these variables. Assuming that the theorem has been proved for the case of 
n — 1 variables, we shall show that it is also true for n variables.. 

From what has been shown in the preceding paragraph, we can always find 
a particular integral uw such that we have U= yu. Replacing one of the inte- 
grals, y, for example, by this integral u, the expressions (58) take the form 


Usain, 
(65) Y, = 0,6 + boo Ya + set DonYny 
. ° ° . . ° . ° ° e 79 
Yn = bp + bn2Yo +++ + OnnYn- 
If in the last n —1 expressions we neglect the terms b,u, +--+, b,u, these equa- 
tions define a linear substitution carried out on the n — 1 variables y,, Y¥3, +++, Yn- 
The determinant D’ of this substitution in n —1 variables is not zero, for the 
determinant D of the linear substitution in n variables is equal to uD’ and can- 
not be zero. Since the theorem is assumed to hold for n — 1 variables, we may 
suppose this auxiliary substitution reduced to the canonical form. This amounts 
to replacing ¥,, Y¥3,+++, Ym by n—1 linearly independent combinations 2z,, Z,, 
+++, Z,—1 such that the equations which define the linear substitution 


Yj = Bin Yg + t+ + Dinn (i = 2, 3, +++, n) 


132 LINEAR DIFFERENTIAL EQUATIONS [ III, § 48 


are replaced by a certain number of groups of equations such as 
Zy = 8%, Zo = 8 (2, + 22), ee 9 Zp = 8(%p—1 + 2p). 


If we carry out the same transformations on the equations (65), it will be 
necessary to add to the right-hand side of the preceding relations terms con- 
taining u asa factor. In other words, we can find n — 1 integrals that form with u 
a fundamental system, and that separate into a certain number of groups such 
that we have for the integrals z,, Z., +++, Zp of a single group 


(66) Z,=sz,+ Kyu, Z,=8(%+2%,)+Kqu, +++, Zp=8(%p-1+%) + Apu, 


where K,, K,,-++, Kp are constants. We shall first try to make as many as 
possible of these coefficients disappear. For this purpose let us put 


U, = 2, +AU, Us = 2 + Agu, ote, Up = Zp + Apu, 


where \,, A», +++, Ap are p constant coefficients. An easy calculation shows that 
we have for these new integrals 


fe = su, + [K, + (x— 8)A\] 4, 


(62) Uh, 2's (iy 1 ee) ER ee 8) ed 


If 1» —s is not zero, we can choose },, ,, +++, Ap in such a way that the coeffi- 
cients of u on the right are zero, and we have for the new integrals u; 


Up — sis, U, =8(u, +t), -**, Up = 8 (Up—1 + Up). 


The substitution to which this group of integrals is subjected after a circuit 
around a is of the canonical form. If w= 8s, since s cannot be zero, we can 
choose ),, \,, +++; Ap—1 in such a way as to make the coefficients of u in the 
expressions for U,, U;,--+, Up, disappear. But we may have several groups of 
variables z, subjected to a transformation of the canonical form for which the 
value of s is equal to w. Suppose, for definiteness, that there are two such 
groups, containing respectively p and q variables. After the preceding change 
of variables the substitutions which these two groups undergo are of the form 


(I) U, = su, + Kyu, U,=S8(ug+ Uy), +7 Up = 8 (Up + Up-1), 
(II) Uj; = su; + Kyu, Ug = 8(ug + U4), a) U; = 8 (U, ot us aa) 


If Ki = K, = 0, we have three groups of integrals, u, (uj, U,, +++, Up)». (Ui, Ug, 

-, Uz), Subjected to a substitution of the canonical form. If we suppose that 
p=q, and if K, is not zero, by putting 1; = ur — K. : ui/K; the second group 
of integrals is replaced by a group of q integrals v, which undergo a substitution 
of the canonical form. Next, putting uy = K,u/s, the (p + 1) integrals uy, u,, 
+++, Uy form a single group which undergoes a transformation of the canonical 
form. If K 1 = 0, while K; is not zero, putting u, = Kju/s, we have two groups 
of integrais, (U1, Uy, +++, Up), (Ug, Uj, +++, Uy), Which undergo a substitution of the 
canonical form. The theorem stated is therefore true in general.* 


*For a full treatment of the application of Weierstrass’s theory of elementary 
divisors to linear differential equations the paper by L. Sauvage (Annales de l’ Ecole 
Normale supérieure, 1891, p. 285) may be consulted. 


IL, § 49] REGULAR INTEGRALS > 133 


49. Formal expressions for the integrals. It remains for us to find a formal 
expression which will show clearly the law of permutation of the integrals of 


the same group after a circuit around the point a. Let y,, y., +++, ¥p» be a group 
of integrals which undergo the permutations (64). Let us put y; = (@ — a)rzZ,, 
where r is equal to Logs/2mi. The p functions z,, 2,--+-, Z must be such 


that we have 

LZ, = 2, Ly = 2; + 2, Sry. Lp = %p—1 + Zp- 
Hence the function z, must be a single-valued function ¢,(x — a) in the neigh- 
borhood of the point a. As to the function z,, we derive from the preceding 
equalities Z,/Z, = z,/z, +1; hence the difference z,/z, — Log(x — a)/277i is a 
single-valued function y,(z — a), and we have also 


Ly pte Log (2% — a) ¢,( — a) + $,(@ — @), 
277i 
where ¢, (2 — a) is another single-valued function. Let us putt = Log (4 — a)/2 wi 
and consider the general case. When a describes a loop in the positive sense 
around the point a, ¢ increases by unity, and z,, 2,,-+-, Zp, considered as func- 
tions of t, must satisfy the relations 


(68) att+l=240, at+l=aQ+aQ, + 
Zp(t + 1) = Zp(t) + %-1(¢). 

In order to find the most general solution of the equations (68), we may 
remark that these relations can be satisfied by taking z, =1, z,=t, and by 
choosing for z,(¢) a polynomial of degree i — 1 in t whose coefficients are deter- 
mined step by step. The calculation is facilitated by observing that the relation 

zi(t + 1) — z(t) = %~1(0) (i = 3) 
is satisfied for ¢=0, 1, 2,---, i— 3 if we take for z,(t) a polynomial of the 
form K,t(¢—1)---({—7i-+ 2). In order that it may be satisfied identically, it 
will suffice if it is satisfied by another value of t, for example, by t = i— 2, since 
the two sides are polynomials of degree i— 2 in t. We thus find the condition 
(i — 1) K; = Ki_1, whence we derive K,;=1/(i—1)!. We therefore obtain a 
particular solution of the equations (68) by putting 
t(t—1)---(E—i+ 2) 

(i—1)! 

In order to obtain the general solution, let us indicate by ¢;(¢) functions such 
that @z(¢ + 1) = ¢z(t). The first of the equations (68) shows that z,(t) is a func- 
tion of this kind, say ¢,(t). The second shows, similarly, that the difference 
z(t) — @,(t)2z,() does not change when we change t tot+1; hence z,(¢) is of 
the form z,(t) = ¢,(t) + 6.¢,(t). We can continue the reasoning step by step. 
Suppose that we have shown that zz, _3(¢) is of the form 


Ze —1(t) = pr—1(t) + A, be—o(t) + +++ + Oe—-19,(L). (k = 3, 4, +++, %) 
The general relation z;(¢ + 1) — z(t) = z;-1(t) shows that the difference 
z(t) — 629i -1(t) — 03 $2 (t) — +--+ — Aidr(t) 
does not change when t changes to t +1; hence the function 2,(¢) is of the form 


Zi(t) = gilt) + 9, Pi-r1(t) + +++ + HO). 


f=; 7,(t) = (i = 2, 3,--+, p) 


134 LINEAR DIFFERENTIAL EQUATIONS [I1I, § 49 


Combining these results, the general solution of the equations (68) is given by 
the relations 
Z(t) = $,(0), 
Z(t) = 4.14) + $2 (4), 
(69) Z(t) = As by (t) + Oo bo(t) + $5(t), 
Zp(t) = Op y(t) + Op—1b9(t) + +++ + Ig bp—ilt) + oh), 
where the functions ¢,, ¢,,-+-+, ¢p do not change when ¢ is changed tot + 1. 
Let us return now to the variable z, and let us indicate by 0,[Log (x — a)]} 
the polynomial in Log (2 — a) obtained by replacing t by Log (x — a)/2 7i in 
6,(t). We see that the p integrals y,, y.,-++, Yp of the group under considera- 
tion, which undergo the substitution (64) after a circuit in the positive sense 
around the point a, are represented by formal expressions of the following type: 


b = («— a)r&,(e — a), 
(70) ]¥2 = @— "18s (Log (@— a} y(e— a) + Bale — a) 
Ip =(t—4)"[Op(Log(v—a)}®,(t—a) + © —1{Log(t— a) }#,(v—a) + ++, 


where ®,(x — a), ®,(% — a), +++, ®p(x — a) are single-valued functions in the 
neighborhood of the point a. 

It will be observed that all the integrals of this group can be deduced from 
the last of them, ¥,, which is of the form 


Yp = (&— a)" [Yo(e — a) + $4 (@ — a) Log(@ — a) + ++ 
+ Pp—1(t — a) {Log (x — a)}?—*], 
where Wo, ¥1,°+*, Yp—1 are single-valued functions in the neighborhood of the 


point a, the last of which, ~,~1, is different from zero.. From the relations (64) 
we have 


_ =P 
Ye Eee eet 


and consequently y,—1 is the product of («— a)" and a polynomial of degree 
p — 2 in Log (x — a), the coefficients of which are single-valued functions in 
the neighborhood of the point a. In the same way we derive yp—» from yp~1, 
and so on. 

If the point a is not an essentially singular point for any of the functions 
©,, &,,--+-+, &,, all the integrals of the group considered (70) are said to be 
regular forz=a. By the remark made on page 131, we can then suppose that 
all the functions ®,(7 — a) are analytic for zg = a, replacing r, if necessary, by 
another exponent which differs from it only by an integer. 


50. Fuchs’ theorem. The determination of the numbers 8,, 8,, +++, 8,, or, what 
amounts to the same thing, the corresponding exponents 7,, r,,++-+, %, is in 
general a very difficult problem. We can obtain these exponents 7; by algebraic 
calculations whenever all the integrals of the equation considered are regular 
in the neighborhood of the point a. This results from an important theorem due 
to Fuchs: In order that the equation (57) shall have n independent integrals, regular 
in the neighborhood of the point a, it is necessary and sufficient that the coefficient 
pi of d"—*y/dx"—* in this equation be of the form (x — a)—‘P;(x), where the func- 
tion P;(x) is analytic in the neighborhood of the point a. 


III, § 50] REGULAR INTEGRALS 135 


If P;(a) is not zero, the point a is a pole of order i for p;; but if P;(a) = 0, 
the point a is a pole of order less than i, It may even happen that the point a 
is an ordinary point for some of the coefficients p;. The preceding conditions 
may be restated as follows: The linear equation must be of the form 
qd y qnr-1 Yy 

7 


Z + (@— a)n-1P,() 5 


+ («—a)P. ~1(0) a oe CAVE BIE 


(71) Sic ras 


where P,, P,,+++, Pn are analytic functions in the neighborhood of the point a. 

We shall develop the proof only for the case of an equation of the second 
order, and we shall suppose, for simplicity, that a = 0. In this particular case 
the first part of Fuchs’ theorem may be stated as follows: Every equation of 
the second order, which has two independent and regular integrals in the neighbor- 
hood of the origin, is of the form 


(72) ay” + cP (x)y’ + Q()y = 9, 
where P(x) and Q(x) are analytic in this neighborhood. 


If the corresponding equation in s (60) has two distinct roots s,, s,, the equa- 
tion (72) has two regular integrals of the form 


(I) Y, = 01¢9,(2), Yo = 22g,(2), 

where the exponents r,, 7, are different and where ¢,(x), ¢,(z) are two analytic 
functions which are not zero for x= 0. If the equation in s has a double root, 
without causing the appearance of logarithmic terms in the expression for the 
general integral, we have again two particular integrals of the preceding form, 
where the difference r, — r,; is an integer. We can always suppose that that 
difference is not zero; for if we had r, = 7,, we could replace y, by the com- 
bination ¢,(0) y,— ¢,(0) y,, which is divisible by 21 **. Finally, if the expression 
for the integral contains a logarithmic term in the neighborhood of the origin, 
we can take a fundamental system of the form 


(11) Y= 219 (Z), Yo = Hb, (a) Log (x) + ¥,()], 

where ¢,(z) is an analytic function which is not zero for x = 0, and where y,(z) 
is a single-valued function in the neighborhood of the origin, which may have 
the point c = 0 for a pole. We have to show that every equation which has two 
independent integrals of the form (I) or of the form (II) in the neighborhood 
of the origin belongs to the Fuchs type. The direct verification does not offer 
any difficulty, but we can abridge the work as follows: If we put y = 2"1¢;(z) u, 
the linear equation in u obtained by this transformation has a general integral 
of one of the forms 


u= 0, + Cx? (x), u= C, + OC, [Log (x) + w(x)], 
where (a) is analytic for « = 0 or has this point for a pole. This equation is 
of the form (72), for the derivative w’ is of the form 
w = Cia" ¢ (2), 


where {(z) is an analytic function which is not zero for g=0. The linear 
ion in u is therefore fe 
equation in wu Is ther up ¢ (2) 


wo (a) 


136 LINEAR DIFFERENTIAL EQUATIONS [III, § 50 


which is of the Fuchs type. Now it is easy to see that this type is preserved 
after a transformation such as y = 21¢,(z)u. The first part of the proposition 
is therefore established. 

In order to prove the converse, let us substitute for y on the left-hand side 
of the equation (72) a development of the form 


(73) Y = Cok? + Car ti .--+ CyartO + o., (Cy # 0) 

and let 
P(t)=a,+a,2+->-, Q(x) = bo + Oo + ++ 
be the developments of the functions P and Q. The coefficient of x” in the 
resulting equation is 
[r(r — 1) + aor + bo] Co. 

Since, by hypothesis, the first coefficient c, is not zero, we must take for r one 
of the roots of the equation of the second degree 


(74) D(r)=r(r—1) + ar+ 6, =0. 


Having taken a root of this equation for r, we can choose c, arbitrarily. Let 
us take, for example, c) =1. Similarly, the coefficient of a +p after the sub- 
stitution is 


el (r + p) (r+ p—1) + a(r +p) +B] + F=epDir +p) +F, 


where F is a polynomial with integral coefficients in Cp, c,, +++, Cp—1, Ay, Ag, ++ *s 
Gp, 0, 0g,+++, bp. Putting successively p = 1, 2, 3,--+-, we shall be able to cal- 
culate, step by step, the successive coefficients ¢,, C,, +++, Cn, unless D(r + p) is 
zero for a positive value of the integer p, that is, unless the equation (74) has a 
second root 7’ equal to the first 7 increased by a positive integer. Discarding 
this case for the moment, we shall obtain a particular integral represented by a 
series of the form (73), the convergence of which will be demonstrated later. 
If the equation D(r) = 0 has two distinct roots.r, 7’, whose difference is not an 
integer, the preceding method enables us to obtain two independent integrals, 
and the general integral is represented in the neighborhood of the origin by the 
expression 


(75) y = C,a"¢ (2) + Chay (2), 


where ¢(x) and y(z) are two analytic functions which do not vanish for « = 0. 

This is no longer the case if the two roots of the equation (74) are equal or 
if their difference is an integer. Let r and r — p be these two roots, where p is 
a positive integer or zero. We can always obtain a first integral of the form 
Y, =v" (x). A second integral y, is given by the general formula (23), which 
becomes here 


dz = “04 a, tagete++) dx 
ta = 2" 6(2) f aoa Ie ) . 


The sum of the roots of the equation (74), or 1— a), is equal in this case to 
. 2r—p; hence a) =p + 1— 2r, and accordingly 


a 
IG @, 4 ae-+*: ‘Jae = gr—(p 4+) §(z), 


where S(x) is a regular function in the neighborhood of the origin, which is not 
zero for x = 0. The second integral y, can therefore be written in the form 


III, § 50] REGULAR INTEGRALS 137 


T (a) dx 
geri 


tn = 29 (0) [ 


where 7'(z) is an analytic function which is not zero for x = 0, 
If A is the coefficient of «? in T(z), we see that the integral y, is of the form 


p1(&) 


LP 


? 


Yo TB (2) |4 Log a + | = a —-Py (x) + Ax’ g(x) Logz, 

where y (x) denotes a new analytic function in the neighborhood of the origin. 
This result agrees precisely with the general theory. As a particular case, it 
may happen that we have A = 0; the general integral does not then contain 
logarithms in the neighborhood of the origin. But since T(0) is not zero, it is 
to be noticed that this case never arises when p = 0, that is, when the equation 
(74) has a double root.* 

To complete the demonstration, it remains only to prove the convergence of 
the series (73) obtained by taking for r a root of the equation (74) such that the 
second root 7’ is not equal to r increased by a positive integer. To simplify the 
proof, we may suppose that r = 0 and that the second root 7’ is not equal to a 
positive integer ; for if we put y = az, the equation analogous to D(r) = 0 for 
the linear equation in z has the roots of the equation (74) reduced by ». We shall 
suppose, therefore, that such a transformation has already been made, so that the 
equation (74) has the root r = 0 and that the second root is not a positive inte- 
ger. For this it is necessary that b) be zero. Modifying the notation somewhat, 
and dividing all the terms by a, we shall write the equation (72) in the form 


(76) ay” + doy’ = wy’ (a, + a0 +++) + y(b, + 0,2 +--+), 


where the coefficients a,, b,, @,,--+ are not the same as before. We are to prove 
that this equation (76) has an analytic integral in the neighborhood of the origin, 
which does not vanish for ¢ = 0, provided that 1— a, is not a positive integer. 
Now, if we try to satisfy this equation formally by a series of the form 


(77) Y=1Lt cot eet cna eee, | 


we obtain successively relations between the coefficients of the form 


(78) Nn {n —1+ ao} = Pr{a,, Ge, °°, B15 bg, ++ +5 On, C1) Coy 2% Cn—Ihy 
(n = il 2, ee -) 


where P,, is a polynomial whose coefficients are all real positive numbers. By 
hypothesis, the coefficient n — 1+ a, does not vanish for any positive integral 


* Let us suppose that the functions P(x) and Q(x) in the equation (72) are even 
functions of x, and that the difference between the roots of D (7) =0 is an odd integer 
2n+1. In this case the logarithmic term always disappears in the integral yg. In 
fact, if we take for the independent variable t=x?, the equation (72) is replaced by 
an equation of the same form, 


(72’) seFU sorts P(ve)] 4 Q(Vt)y=0, 


and the roots of the equation analogous to D (r)=0 are, as is easily verified, half of 
the roots of D(r)=0. Since their difference is not an integer, it follows that the 
general integral of the equation (72’) does not contain any logarithmic term in the 
neighborhood of the origin. The same thing is therefore true of the equation (72). 


138 LINEAR DIFFERENTIAL EQUATIONS (III, § 50 


value of n. We can therefore determine a positive number yu such that we have, 
for every positive integral value of n, |n —1+ a)|>(n + 1), since the quotient 
(n —1-+ a,)/(n + 1) approaches unity as n becomes infinite. Let us replace, on 
the other hand, the coefficients of zy’ and y on the right-hand side of the equa- 
tion (76) by dominant functions, and let us consider the auxiliary equation 


(79) e(aY”’ +2Y)=2Y(Ai+ Ae +--+) + Y(B, +Bet+-->). 


If we attempt to satisfy this new equation by a series of the form 


(80) Y=14+C,e+ ---+C,a"+>>-, 
we are led to the relations analogous to the relations (78), 
(81) nun (n A 1)= Pr(Ay, A,, ene B,, B,, rage? C1, s+) On—1)- 


If we compare the expressions which give the values of the coefficients c, and Cy, 


i: P,(a,, Gg, ***s b,, b, Vitek Ce Ca—1) gh mae P,(A,, Aes tags, Cp —1) 
n(n —1+ a) n(n + 1) 


the conditions 4;= |a;|, B;=|b;|,|r —1+ a)|=u(n + 1) show successively that 


Cn 


[ce |<C,, [Ce] <1 Cs, gt NAL, Lt Fae 


hence it will suffice to show the convergence of the auxiliary series or to show 
that the equation (79) has an analytic integral, in the neighborhood of the origin, 
not vanishing for z= 0. If we take for the dominant functions an expression 
of the form M/(1— @/r), the auxiliary equation (79) can be written 

CL 2 YU SMA 


b] 


eY’+Y be 1-2 
" 
whence we derive, by a first integration, 
Mr 


cY/ ay o(1- =) “ 
‘ r 
and then 


Mr 
ey =0 f(1—*) * dt + OC’. 
0 Dy 


We have only to take C’ = 0, C = 1 in order to have an analytic integral, in the 
neighborhood of the origin, not vanishing for z = 0. 

Extension to the general case. The proof of Fuchs’ theorem for the general 
case can be based on the same principles by showing that if it is true for an 
equation of order (n — 1), it is also true for an equation of order n. 

If the equation (57) has n particular integrals separating into a certain num- 
ber of groups of the form (70), it has at least one particular integral of the form 
(x — a)" p(x — a), where ¢(x — a) is an analytic function in the neighborhood 
of the point a, which does not vanish for z = a. The substitution 


y =(r&— a)"po(x—a)u 
will lead to a linear equation in u which has the particular integral u=1; 
hence the derivative w’ satisfies a linear homogeneous equation of order n — 1. 
The theorem being supposed true for a linear equation of order n—1, this 


equation in w’ is of the Fuchs form; the same thing is evidently true of the 
equation in u and therefore of the equation in y. 


III, § 50] REGULAR INTEGRALS 139 


Conversely, let us consider an equation of the form (71), in which a= 0. 
This equation is formally satisfied by a series of the form 


Y= Cot” + cyarti4+..., (Cy 4 0) 
where r denotes a root of the fundamental characteristic equation 
(82) Masi kate 
+ P,(0)r(r—1)---(7r—n+ 2)4+ +--+ P, (0) =0 


such that no other root of this same equation is equal to r increased by a posi- 
tive integer. In order to establish the convergence of this series, it is easy to 
show, by an artifice analogous to the one employed for n = 2, that it suffices to 
prove that a linear equation of the form 


qn 
rp ty) = 


has an analytic integral in the neighborhood of the origin not vanishing for 
x= 0. Now this equation has the particular integral (§§ 18 and 39) 


— Mr 
ne oe dt, 
Uf Se Py dares 0 


which actually satisfies the preceding condition. If the equation (82) has n 
distinct roots, rj, 7), +--+, 7, Such that none of the differences r; — rz is equal to 
an integer, the general integral of the linear equation is of the form | 
y = OC, 2719, (€) + CL 22 Gy (LZ) + +++ + Cr@ndGn(Z), 

where $1, $5, °°*; $n are analytic in the neighborhood of the origin. If the 
equation (82) has equal roots or, more generally, roots such that some of the 
differences 7, — rz are integers, these roots separate into a certain number of 
groups, the difference between two roots of the same group being an integer, 
while the difference between two roots of different groups is never an integer. 
Let r be the largest root of one of these groups. We have just seen that the 
equation (71) has a particular integral of the form x" ¢(z), where ¢(z) is an 
analytic function in the neighborhood of the origin and such that ¢ (0) is not 
zero. By putting y = 2" ¢(z)u, then du/dz = v, we are led to a linear differ- 
ential equation of order n —1 in v, which is again of the Fuchs form. The 
theorem being supposed true for an equation of order n — 1, that equation in v 
has n — 1 particular independent integrals of the form 


v = £7 [Yo (£) + , (2) Loge + «++ + q(x) (Log x)2], 
where yp, ¥1,°°+, Wq are analytic functions for = 0. If @ is not an integer, 
we easily see, by a succession of integrations by parts, that 44 vdz is an expres- 
sion of the same kind as v. If a is an integer, f{vdz containsalso a logarithmic 
term 
C (Log x)¢+1, 

where C is a constant coefficient. Fuchs’ theorem is therefore true for an 
equation of the nth order.* 


* For greater detail see the paper by Fuchs in Crelle’s Journal or the thesis of 
Jules Tannery (Annales de l’ Ecole Normale, 2d series, Vol. IV, 1875). 


140 LINEAR DIFFERENTIAL EQUATIONS [III, § 51 


51. Gauss’s equation. Let us apply the general method to the equation 

(83) e(l1—2x)y” + [y—(@+B84+1)r]y’ — aby =), 
where a, 8, y are constants. The singular points in the finite plane are z= 0 
and «=1. The characteristic equation for the point z = 0 is r(r+ y—1)=90, 
and its roots are r=0, r=1—y. If y is not zero nor equal to a negative 
integer, it follows from the preceding theory that the equation has an analytic 
integral in the neighborhood of the origin corresponding to the root r= 0, In 
order to determine this integral, let us substitute in the equation the series 


Y =Co ACYL + e+ +Cnt" + «-- 


and equate to zero the coefficient of a*-1. This gives a recurrent relation 
between any two consecutive coefficients 


n(ytn—l1)eq, =(a+n—1)(8+n—1)e,-1; 
hence the analytic integral is the series 


aa+) BB+) oy... 
1.2.¥(y +1) ; 


which is called the hypergeometric series. This series is convergent in the circle Ty 
with unit radius about the origin as center. In order to obtain a second integral, 
let us make the transformation y = 21-Yz. This leads to an equation of the 
same form, 
(84) 1 Et Es CR ea ae 
— (a bly) (BP ay) 2 0, 


which differs from the first only in the substitution of a+1—y,B+1—y,2—y 
for a, B, y respectively. If 2—y is not zero nor equal to a negative integer, 
the equation (83) has therefore the second integral z!1-Y F(a+1—y, B+1—y7, 
2— +, x); and if y is not an integer, the general integral is represented in the 
circle I'y by the expression 


(85) y=C, F(a, B, y, 2) + C,a-YF(a+1—y, B+1—y, 2—7, 2). 
If y-is an integer, the difference between two roots of the characteristic 
equation is zero or equal to an integer, and the integral contains in general 


a logarithmic term in the neighborhood of the origin. We shall study only the 
case where y =1. The two integrals 

F(a, B, ¥, £), BE UE (Ody Bed iy, torre ee) 
reduce in this case to the single integral F(a, B, 1, 2). 

In order to find a second integral, let us first suppose that y differs but little 
from unity, say y =1—h, where hf is very small ; then the equation (88) has the 
two integrals 

F(a, p,1-h, 2), we F(ath, B+h,1+h, a), 
and consequently the quotient 


aF(ath, B+h,1+h, x)— F(a, B, 1—h, x) 
h ; 
is also an integral. As h approaches zero, this quotient approaches as a limit the 
derivative of the numerator with respect to h at the point h=0. The deriva- 
tive of the factor z* gives us a logarithmic term which, for h = 0, reduces to 
F(a, 8,1, 2)Log@. To find the derivative with respect to A of any coefficient 
in the two series, such as the coefficient 


\ 


Dey 


III, § 51] REGULAR INTEGRALS 141 


(a+ h)(@+h4+1)---(@+h4n—1)(B+h)(B4+h41)---(B+h4+n—1) 
n! (1+ hA)(2+A)...(n+ A) 
it is convenient to calculate first the logarithmic derivative. We find thus a 
new integral which has the form 


¥,(c) = F(a, 8, 1, x) Loge 


(86) 4 eA, Ma+ I++ (at n— 1A +1)---G+n—1) 
(n !)? 
where we put 
1 1 t 1 1 
ee ae 3g een are a eee fe ee Meme 
Ra ere al PET ip gn a as raat 


We might. study in the same way the integrals of Gauss’s equation in the 
neighborhood of the point x=1, but it suffices simply to notice that if we 
replace x by 1— a, the equation does not change in form, but y is replaced by 
a+ 8+1—y. The general integral is therefore represented in the circle I, 
with unit radius about the point x =1 as center by the expression 


y= OC, F(a, Bp, a+ B+1—y,1—2) 
nt 5 (Bie) Cae BH (yr, ¥— 6) y + l— @— Bp, 1— 2x), 


provided that y — a — @ is not an integer. 

In order to study the integrals for values of x of very large absolute value, 
we put «=1/t, and we are then led to study the integrals of a new linear 
equation in the neighborhood of the origin. The integrals of this equation 
are likewise regular in the neighborhood of the origin, and the roots of the 
characteristic equation are precisely aw and B. If we substitute simultaneously 
x=1/t, y=t*z, the equation obtained is again of the form (83), but @ is 
replaced by a+1—y, and y by a+1— 8. Gauss’s equation has therefore 
the integral 


r#F (a, a+l—y,a+1-8, *). 
x 
By symmetry it has also the integral obtained from this one by interchanging 


« and f, and therefore the general integral is represented in the region exterior 
to the circle I, by the expression 


1 1 
y=Cy2-«F (a, a+tl—y,at+ 1-6.) +C,0-8F (8+ 1—y, 6, B+ 1—a,=), 
provided that a — @ is not an integer. 

Note. Every linear equation of the form 


(87) (a — a) (a — b)y” + (lw + m)y’ + ny = 0, 


where a, b, l, m, nare any constants (a ~ 6b), reduces to Gauss’s equation by the 
change of variable zr =a+(b—a)t. For, to identify the resulting equation 


d?y la +m \¢ 
88 EO ae to) — — —0 
font — ) on Ga rr tess 


with the equation (83), we need only put y =— (la + m)/(b — a), and then 
determine @ and f by the two conditions a+B+1=1, aB=n. 


142 LINEAR DIFFERENTIAL EQUATIONS [III, § 52 


52. Bessel’s equation. Let us consider in particular the equation 
(89) x(1— kxr)y”’ + (c — bxr)y’ — ay = 0,7 


which has the two singular points z = 0, 7 = 1/k, and which can be reduced to 
Gauss’s equation by the change of variable kx = t. If we make the parameter k 
approach zero while a, b, c approach finite limits A, B, C, the singular point 
x = 1/k goes off to infinity, and we obtain at the limit the linear equation 


(90) ry’ + (C— Br)y’—Ay =0, 


whose only singular point at a finite distance is the pointz=0. If B is not 
zero, replacing Br by x we are led to an equation of the same form, where 
B=1. Likewise, if B =0 and 4 is different from zero, we can suppose A = 1. 
Finally, disregarding the trivial case 4 = B=0, the equation (90) can be 
replaced by one of the two forms 


(91) ny” + (y—2)y' — ay =0, 
(92) ay” + yy’ —y =0. 


Studying the integrals of these two equations in the neighborhood of the 
origin, as we have done for Gauss’s equation, we are led to introduce the two 


series a(a +1) a3 -y 

1.2.y¥(74+1) ? 
1 ay 

1.2.y(74+1) : 


which may be considered as degenerate cases of the hypergeometric series. If 
we replace in F(a, B, y, x) the variable x by kx and B by 1/k, the coefficient 
of a in F(a, 1/k, y, kx) approaches the coefficient of 2 in G(a, y, x) as k& 
approaches zero. Similarly, the coefficient of a in F'(1/k, 1/k, y, k?x) approaches 
the coefficient of 2” in J (y, x) as k approaches zero. 

If y is not an integer, the general integral of the equation (91) is given by 
the expression 


(93) y= C1G (a, , ©) + C,01-YG(a+1—y7, 2—¥, 2). 


G(a, y,2)=14——a4+ 
Ly 


1 
TORE red a te 


Likewise, the general integral of the equation (92) is 
(94) y =O, (y, ©) + C,x21-YJ (2 — ¥, x). 


These formule are valid in the whole plane. 

If y is an integer, the general integral of the equation (92) always contains a 
logarithmic term. For example, if y =1, we obtain an integral different from 
J(1, x) by finding the limit for h = 0 of the quotient 


rI(1+h,2)—J(1—A, 2) 
h mt A 
which gives for the general integral 


+ 
1 1\ a 
= C,J(1, x) + C,| J(A, Loge — 25'(14+= -++ + —)—_]}. 
Mae a Nie) st [7 Tove : ( aoe: Salen, 


We can reduce to the form (92) a certain linear equation which appears 
in a large number of questions of mathematical physics, Let us put in the 


III, § 53] REGULAR INTEGRALS 143 


equation (92) « =— t?/4; replacing y by n+ 1, the equation obtained is iden- 
tical with the equation already studied (§ 46), 


d?y dy 
95 t—= 4+ (2n+1)—+ty=0. 
(95) gat Chaat 


If, in this last equation, we put y =t-"z, we obtain a new form of Bessel’s 


equation, 
G22. as 
96 t?_ 4 t¢ — + (1? — n?)z= 0. 
(96) Hane aris ) 
The three equations (92), (95), (96), where y = n + 1, are therefore absolutely 
equivalent to one another. If n is not an integer, the preceding development 
shows that the general integral of Bessel’s equation (96) is 


ti? {2 


We have shown above (§ 46) that if n is half an odd integer, the general integral 
of the equation (95) can be expressed in terms of elementary transcendental 
functions. Hence the transcendental function J (y, x) is expressible in terms of 
exponential functions if y is half of an odd integer. 


Note. The equation studied by Riccati, 


(97) Le: + Au? — Ber = 0, 
dx 


where A, B, m are given constants, can also be reduced to any one of the 
equivalent equations (92), (95), (96). Indeed, we have seen (§ 40) that the gen- 
eral integral of the equation (97) is z’/Az, where z is the general integral of 
the linear equation 

(98) rae A But z'= 0. 

dx? 

If we make the change of variable « = dt“, where \ and yw are two undetermined 
quantities, the last equation becomes 

(99) 1S — u— 1) 92 — ABnm42patin DH Iz = 0. 
In order to identify this equation with the equation (95), we need only take 
= 2/(m + 2), and determine X by the condition ABA™+2y?2 =—1. The cor- 
responding value of n is — 4/2 or —1/(m + 2). We can therefore express the 
general integral of Riccati’s equation (97) in finite terms whenever 1/(m + 2) 
is half of a positive or negative odd integer 27 + 1, that is, whenever m is equal 
to — 4i/(1+ 2%), where i denotes a positive or negative integer. 


53. Picard’s equations. Given a linear differential equation with coefficients 
analytic except for poles, we can determine by Fuchs’ method whether the 
general integral is itself an analytic function except for poles. For this it is 
necessary and sufficient: (1) that the integrals shall be regular in the neigh- 
borhood of each of the singular points; (2) that all the roots of the charac- 
teristic equation, relative to each of these singular points, shall be integers ; 
finally, (8) that all the logarithmic terms shall disappear from the expression 
for the general integral in the neighborhood of a singular point. 


144 LINEAR DIFFERENTIAL EQUATIONS [III, § 53 


Suppose that all these conditions are satisfied. The general integral is then 
a single-valued analytic function except for poles in the whole plane. If the 
coefficients of the equation are rational functions, there are only a finite num- 
ber of singular points a,, @,,-++, @,. In order for the general integral to be a 
rational function, it is sufficient that the equation obtained by putting x = 1/t 
shall itself have all its integrals regular in the neighborhood of the point ¢ = 0, 
since the general integral is single-valued and therefore cannot contain log- 
arithmic terms nor fractional powers of t. If this last condition is satisfied, we 
can obtain the general integral by equating coefficients according to the method 
of undetermined coefficients. In fact, let — m; be the smallest root of the char- 
acteristic equation relative to the point x = a;, and WN the smallest root of the 
characteristic equation relative to the point t = 0 for the transformed equation. 
It is clear that the product of any integral y and the expression 


(@ — dy) (t= a) 2+ + + (@— ay) 


is a rational function having no poles in the finite portion of the plane. This 
product is therefore a polynomial P(x), whose degree is at most equal to 


M,+m,+---+m,—N. 


Since we know an upper bound for the degree of this polynomial, the coefficients 
can be determined by replacing y by an expression of the form P(z) II (x — a;)—™, 
where P(z) is the most general polynomial of this degree, in the left-hand side 
of the given equation, and then equating the result identically to zero. 

Picard has given another very important case where the general integral can 
be expressed in terms of the classic transcendental functions. Given a linear 
homogeneous differential equation, whose coefficients are elliptic functions of the 
independent variable with identical periods, if its general integral is an analytic 
function. except for poles, that integral can be expressed in terms of the standard 
transcendental functions of the theory of elliptic functions. 

For simplicity in writing, let us develop the proof for an equation of the 
second order only. Let f,(x), f,(z) be two independent integrals of a linear 
homogeneous equation y” + p(x) y + ¢(x)y = 0, where p (z) and q (2) are elliptic 
functions with the periods 2 and 2’. By hypothesis, f(x) and f,(x) are single- 
valued functions analytic except for poles. Since the given equation does not 
change when we replace « by © + 2w, f(x + 2w) and f,(x + 2w) are also inte- 
grals, and we have the relations 


(100) F(a + 2) = af, (x) +0f,(2), F,(@ + 2w) = cf,(x) + af, (2), - 


where a, 0, c, d are constant coefficients whose determinant ad — bc is not zero. 
For if we had ad—bc=0, we could derive from (100) a relation between f, («+2 w) 
and f,(% + 2w) of the form C,f,(x + 2w) + C,f,(« + 2) = 0, where C, and C, 
are constants not both equal to zero. This is impossible, since f, and f, are two 
independent integrals. For the same reason, we have another system of relations 


(101) f,(@ + 20’) = af,(x) + O4(z), J,(@ + 2w’) = cf, (x) + df, (2), 


where a’, b’, c’, d’ are constant coefficients, and a’d’ — b’c’ is not zero, Let us try 
to find, as in § 47, an integral ¢ (x) = Af, (x) + wf,() such that ¢(@ + 2 w) = sp(z). 
We have for the determination of A, u, s the two equations 


A(a— 8) + wco= 0, b+ u(d—s)=0; 


III, § 53] REGULAR INTEGRALS 145 


whence we derive the equation of the second degree for s, 
F(s) = s?*— (a+ d)s + ad— be = 0. 


If this equation has two distinct roots s,, 8,, there exist two independent inte- 
grals $,(x), ¢2(x) such that we have 


(102) $,(e + 2w) = 8,6, (2), $o(& + 2w) = 8,4,(Z), 


and the relations (101) can be replaced by the two relations of the same form 


(103) $,(@ + 2’) = kd, (x) + Ip(x), a(t + 2 w’) = mg, (x) + ngp(z). 


By means of the relations (102) and (103), we can now obtain two different ex- 
pressions for ¢,(% + 2w + 2’) and ¢,(% + 2w + 2w’). We have, on the one hand, 


$4(€ + 2w + 2w’) = 819,(€ + 20’) = 8,kh,(£) + 8,/6,(2). 
On the other hand, proceeding in the inverse order, we may also write 
$i(€ + 20 + 2w’) = kp, (H+ 2w) + lpy(w + 2 w) = ks, 9, (x) + 1s8,h,(c). 


Since these two expressions must be identical, we have |= 0, for s, — s, is not 
zero. Similarly, by considering the two expressions for ¢,(% + 2w + 2w’), we 
find m=0. The integrals ¢,(x), ¢,(x) are therefore analytic functions except 
for poles, which reproduce themselves multiplied by a constant factor when the 
variable x increases by a period ; these are called doubly periodic functions of the 
second kind. Every function ¢(zx) analytic except for poles which possesses this 
property can be expressed in terms of the transcendental functions p, ¢, o, since 
the logarithmic derivative ¢’(x)/¢() is an elliptic function, and we have seen 
that the integration does not introduce any new transcendental (II, Part I, § 75). 
Moreover, we can prove this without any integration. Let ¢(z) be an analytic 
function except for poles such that 


P@+20)=nP(e), P(e+2w’)=p’'G(a). 


Consider the auxiliary function y (x) = eP*o (x — a)/o (x), where a and p are 
any two constants. From the properties of the function ¢ (see Vol. II, Part I, 
§ 72) we have 


(t+ 2) = 2ep—270y (2), (w+ Qu’) = w’P—2"ay (2), 
In order for the quotient ¢ (x) /y (2) to be an elliptic function, it is sufficient that 
2wp — 2an = Log yp, 2w’p — 2an’ = Log p’. 


These relations determine p and a (II, Part I, p. 161). It should be noticed that 
we can take a= 0 if Loguw and Logw’ are proportional to the corresponding 
periods 2 w, 2w’. 

Let us now turn to the case where the equation F'(s) = 0 has a double root s. 
We can find (§ 48) two independent integrals ¢,(z), ¢,(z) such that 


(104) (w+ 2u)=s9,(2), g(t + 2w) = 8992) + CO, (2). 


If C = 0, all the integrals of the equation, and in particular f,(x) and f,(z), are 
multiplied by s when z is increased by 2w. Assuming C = 0, let us try to find a 
linear combination Af, (x) + »f,(x) which reproduces itself multiplied by s’ when 


146 LINEAR DIFFERENTIAL EQUATIONS [III, § 53 


x increases by 2w’. Starting from the equations (103), we find two independ- 
ent integrals ¢,(x), ¢.(x) such that either 


(t+ 20) =81b,(2), — bp(% + 2’) = 8344(z) 
or 
$1(& + 2’) = 8’9,(2), o(& + 2’) = 8’$,(x) + C’g, (Zz), 

where ©’ is not zero. In the first case the integrals ¢,(z), ¢,(x) are again 
doubly periodic functions of the second kind. In the second case the integral 
¢,(v) alone is a doubly periodic function of the second kind. As for the inte- 
gral ¢,(x), the quotient ¢,(x)/¢,() increases by a constant C’ when « increases 
by 2w’, and it does not change when @ increases by 2w. Now the function 
A(x) + Bx, where A and B are two constant coefficients, possesses the same 
property, provided that we have 


2An+ 2 Bw = 0, 2 An’ + 2 Bw’ = C’. 


The difference $,/¢, — A¢(x) — Bx is therefore an elliptic function. 

If the coefficient C is not zero in the equations (104), we have relations between 
the integrals $,(£), ¢.(), $)(© + 2’), (© + 2w’) of the form (103), and we 
can again deduce from them two different expressions for ¢,(% + 2 + 2w’) 
and ¢,(« + 2w+ 2’). By writing that they are identical, we obtain the con- 
ditions 1=0,k=n. The integral ,(2) is again a doubly periodic function of 
the second kind, while the integral ¢, (x) satisfies the two relations 


GRE eld boo Sd SUAS Go(@ +20’) _ $2 (x) m- 
(+2) (@%) 8 (+20) $() & 
Let us determine just as before the two coefficients A and B in such a way 
that 2An + 2 Bw = C/s, 2 An’ + 2 Bw’ = m/k. Then the difference 


G20) Ae(e) — Be 

$; (2) 
is again an elliptic function. We see, therefore, that the general integral is in 
all cases expressible in terms of the single transcendentals e*, p(x), ¢(z), o (a). 

Let us consider, for example, Lamé’s equation 

d*y 
da? 
where n is an integer and hf is an arbitrary constant. The integration of this 
equation by Hermite was the starting point for the preceding theory. The gen- 
eral integral of this equation is a function analytic except for poles. In fact, 
the only singular points are the origin and the points 2mw + 2m’w’. In the 
neighborhood of the origin the integrals are regular, and the roots of the char- 
acteristic equation are 7” =—n, r’ =n+1. Their difference is an odd integer, 
and the coefficient of y is an even function; therefore the expression for the 
general integral does not contain any logarithmic term (see ftn., p. 187). 


(105) — [n(v+1)p(e) + h]y=9, 


54. Equations with periodic coefficients. In many important questions of 
mechanics, linear equations with periodic coefficients occur. We shall indicate 
rapidly their more important properties. Let 


dry dn-ly 


106 
Wet din "73 qi 


+ +++ + pny =0 


Il, § 54] REGULAR INTEGRALS 147 


be a linear equation whose coefficients are continuous functions of the real 
variable ¢t, having a period w, which we may always suppose positive. If the 
integrals y,(t), y2(4),---, Yn() form a fundamental system, it is clear that 
Yx(E+ w), Yo(t+ ),--+, Yr(6 +) are also integrals of the equation (106), 
since that equation remains unchanged when we replace t by t+ w. Hence 
we have n relations of the form 


(107) Yi(t + w) = ai y1(0) + Ajieye(t) + +++ + AinYn (Et). (i =1, 2,---, n) 


The determinant H of the coefficients az is different from zero. For, by 
repeating the reasoning of page 129, we find that this determinant has the value 


ee 
(108) Het, 


The equations (107) define a linear substitution with constant coefficients, 
whose determinant is not zero. We are therefore led to a study entirely similar 
to the one which has already been made in detail in §§ 48, 49. Instead of 
making the complex variable x describe a circuit in the positive sense around a 
singular point a, the variable t describes a segment of the real axis of length w. 
It follows from that study that we can always choose a fundamental system of 
integrals such that the relations (107) reduce to a simple canonical form. The 
actual formation of this system depends first of all on the solution of the 
characteristic equation 


a1 — 8 a2 SNC Ain 
a21 d22 — 8 a, a2 
(109) F(s) = ae bs == ()3 
Ani On2 2 ce Ann — § 


All the roots’ of this equation are different from zero, since their product is 
equal to the determinant H, whose value we have just written down. If the 
n roots of that equation are distinct, there exists a fundamental system of 
integrals such that the equations (107) take the form 


(110) y(t + w) = 8, Y, (0), +++, Yn(L+ &) = SnYn(t). 


If the equation (109) has multiple roots, we can always find a fundamental 
system of integrals which separate into a certain number of groups such that 
the p integrals ¥,, Y¥,-+++, Yp of the same group satisfy relations of the form 


Y(t + w) = sy, (¢), 
Yp (t + w) = 8[¥p (t) + Yp-1(4)]. 

In order to find expressions for these integrals, let us seek first the general 
form of a single-valued continuous function f(t) such that f(t + w) = sf(6), 
where the factor s is not zero. Let a be a determination of (1/w) Logs. It is 
clear that the product f(t)e-%* has the period w; hence f(t) is of the form 
ST (t) = et* (t), where ¢(¢) is a continuous function with the period w. Accord- 
ingly, if s; is a root of the characteristic equation, we shall put a; = (1/w) Log s;. 
The constants a@;, which are determined except for multiples of 24 V—1/w, 
are called the characteristic exponents. The real parts of these exponents, which 
are determined without ambiguity, are called the characteristic numbers. If the 


148 LINEAR DIFFERENTIAL EQUATIONS [IIT, § 54 


equation (109) has n distinct roots 8,, 8,, +++, S:, the equation (106) has then n 
independent particular integrals of the form 


(112) yy = et’, (t), Yo = €%'d, (lt), rays Yn = e%n* d, (t), 
where @, @,+++, & are the characteristic exponents, and where $,, $5, +++, $n 
are continuous functions with the period w. 

In the general case it is evidently sufficient to find expressions for the 


integrals of a group which satisfy the relations (111). Now if we substitute in 
these relations y; = e**z;, where @ is equal to (1/w) Log s, they become 


Z(t+ w)=2,(), 

(111/) Z(t + w) = Z(t) + 2, (4), 
split a) = ap) + tp). 

When ¢ increases by w, the variable t = t/w increases by unity. Taking rt for a 

new variable, the problem is reduced to one solved above (§ 49). If we set 


HS wear (Ce ee) 


wi! 


Bho = (t= 1, 2,---, p) 


the general expressions for the functions y,, Y¥.,+++, Yp are 


YW =e P(t), Y=" PIH HOH GO], ++ 
(118) < yi(t) =e [Pi-1(€) O (6) + Pi—2 (Ct) G2 (t) +--+ + PO Gi-1) + Gi (I; 
(i =1, 2, ++, p) 


where ¢$,, $5;° eh, dp are continuous functions with the period w, the first of 
which, ¢,(¢), is not zero. We see again here, as in § 49, that all these integrals 
can be deduced from the last one of the group. For Z)~_1(t) is equal to the 
difference z, (¢ + w) — Z(t), and, similarly, z, 9 (¢) = Z%-1(t + ) — Zp-1(0), and 
so on. We can therefore write the equations (113) in the form 


Y(t) = e*Zy(t), 
Yp 1 (0) = er A, (Zp), 
(114) Yp —2(t) = e Ay (Zp), 

y(t) =e Ap-1(%), 
where A,(Zp), A, (Zp), +++ indicate the successive differences of z,(t) when we 
change t tot+w. Let us observe that z, (t) is a polynomial in t of degree p—1, 
whose coefficients are periodic functions of t. The successive differences A, (zp), 
A, (Zp), +++ are therefore polynomials of the same kind with decreasing degrees, 
the pth difference being zero. Let us indicate by Dz,, D*Zp,-+-, D*zp the suc- 
cessive derivatives of z, taken with respect to t, considering the coefficients of 
this polynomial as constants. From the theory of finite differences, we know 


that the successive differences A,(Z»), A, (Zp), +++ are linear combinations with 
numerical coefficients of the derivatives Dz), D*z,,---, D'zp, and conversely.* 


* Without resorting to this theory, we may observe that Taylor’s formula gives us, 


step by step, Ai (Zp) = Wt Dizp +--+, 


where the terms not written contain only the derivatives Dit+1,---. We can there- 


fore express, conversely, the derivatives D‘zp as linear functions of the differences 
Ai, TANS sh bs eg rs 


att 


II, § 54] REGULAR INTEGRALS 149 


‘We can therefore replace the system of integrals (114) by the equivalent system 


Yp(t) = e%Zp(t), 

Yn—1(t) =e Dz, 
(115) Yp—2(t) = e4% D*z,, 

Yee (= Dea. 

Note 1. The integrals of the group (113), corresponding to the characteristic 
exponent a, approach zero when ¢ becomes infinite passing through positive 
values, if and only if the real part of a@ is negative. In order that all the inte- 
grals of the equation (106) shall approach zero as t becomes infinite, it is there- 
fore necessary and sufficient that all the characteristic numbers shall be negative, 


or, what amounts to the same thing, that the absolute value of each of the roots 
of the equation (109) is less than unity. 


Note 2. If s is a real positive root of the equation (109), it is natural to 
take for @ the real determination of (1/w) logs. If the coefficients of the equa- 
tion (106) are feat the same thing will evidently be true in this case of the 
integrals ¥,, Yo,+**; Yp Of the group (113) and consequently of the periodic 
functions ¢,(¢), PY (t), ++ 

Lets =A + "4V—1 bea p-fold root of the equation (109), where » ~ 0, and 
let a= a’ + a&” V—1 be a corresponding determination of the exponent a. To 
the group of integrals (118) we can adjoin a conjugate group obtained by replac- 
ing @ by a — a’ V—1 and the functions ¢; (t) by the conjugate functions. It is 
clear that by combining these 2 p integrals linearly in pairs we can derive from 
them a system of 2p real integrals. 

Finally, suppose that s is a real negative root. Then we can write the value 
of a= a’ + (7/w) v=1, and to that root corresponds a particular integral of 
the form 


y= ex’t (cos = + V1 sin =) CA (t) + Veet y, (t)), 


where the functions y, and y, are real and periodic. If the coefficients of (106) 
are real, it is clear that the real part and the coefficient of /—1 must each 
satisfy separately the linear equation. We would proceed similarly with the 
other integrals of the group (118) if p is greater than unity. 

Moreover, the case where s is real and negative reduces to the case where s 
is real and positive by considering the period 2w instead of the period w. It is 
clear, in fact, that if an integral is multiplied by s when we change ¢ to t + wo, 
it will be multiplied by s? when we change ¢ to t + 2w. 


Note 3. When the coefficients p; are analytic functions of the complex vari- 
ablet =U +t” V ith analytic in the strip R included between the two parallels 
to the real axis ’” =+ h, the integrals of the equation (106) are analytic func- 
tions in the same strip. The reasoning used under the supposition that the 
variable t moves along the real axis applies without modification to the case 
in which that variable moves in the strip R. It follows that the functions ¢;(¢), 
which appear in the expressions of (118), are periodic analytic functions in the 
strip R. They can therefore be developed in series of sines and cosines of 
multiples of the angle 2 wt/w (see Vol. II, Part I, § 65). 


150 LINEAR DIFFERENTIAL EQUATIONS [III, § 55 


55. Characteristic exponents. The investigation of the characteristic exponents 
is in general very difficult.* The solution of this problem evidently reduces to 
the determination of the coefficients a, which appear in the equations (107), 
which, in turn, is equivalent to the following: knowing the initial values, for 
t =t,, of the n integrals y,, Y2,--++, Yn and their first n — 1 derivatives, to find 
the values of these integrals and of their derivatives for t = t), + w. The coeffi- 
cients a; are then obtained by the solution of the n systems of linear equations 


Yi (to + w) = a1 Y1 (to) + AiaYe (bo) + +++ + Gin Yn (to) 
(116) 4 of? lo + w) = ain yh?” (lo) + = + ain? 
(p =1, 2,...,(~—1)) (@=1, 2,..., n) 


We cannot in general solve this last problem except by the use of general 
methods, for example, by successive approximations. Let us replace p,; by Ap; in 
the equation (106), where \ denotes a variable parameter, and then develop in 
powers of \ the integral of that equation which together with its first (n — 1) 
derivatives takes on preassigned values independent of d for t = ft), 


(117) Y =Sy(t) + M+ es FMA Fos 


where f, (t) is a polynomial in t, of degree n —1 at most, which can be written 
down immediately from the initial conditions. Substituting this value of y in 
(106), we see that the other coefficients /, (d), f, (d), -- - are determined, step by step, 
by relations of the form 


nT. 

SF = Olt, AO KOs 
in which the right-hand sides depend only upon the functions f,, f,, +++, fi-1, 
and upon their derivatives. Moreover, these coefficients, together with their first 
n — 1 derivatives, must vanish fort =¢). Hence these coefficients can be found 
by quadratures. We have already noticed (§ 28) that the series obtained is con- 
vergent for any value \. If we put \ = 1 in the relation (117) and in all those. 
which we obtain from it by differentiation, we shall have the developments of 
the integral under consideration and of its derivatives in series which are con- 
vergent for all real values of ¢. Hence we can obtain in this way the quantities 
Vr (ty + w), y§” (tp + w) which appear in the equations (116), and consequently 
we can determine the coefficients a;,.t 


Example. Let us consider, for example, the equation 


; d2y 
(118) diz = p(y, 
where p(t) is a continuous function of t with the period w. The product of 


the roots of the characteristic equation is here equal to one, by formula (108). 


* When the coefficients pi are analytic integral functions of the complex variable ¢, 
the change of variable e27t/»=% replaces the given equation by a linear equation 
whose coefficients are single-valued in the neighborhood of the origin, and we are led 
to study the law of the permutation of the integrals when the variable x describes 
a loop around the origin. But the equation thus obtained is not in general of the 
Fuchs form. 

t If we allow the parameter \ to have any value, it follows, from the process used 
above, that the coefficients aiz, and consequently the coefficients of the characteristic 
equation, are integral functions of this parameter. 


III, § 55] REGULAR INTEGRALS 151 


That equation is therefore of the form 
(119) s?—As+1=0. 
In order to determine the coefficient A, let us denote by f(t) and ¢(t) the inte- 
grals of the equation (118) which satisfy the initial conditions f(0) = 1, /’ (0) =0, 
¢ (0) = 0, ¢’(0) =1. From the relations 
mA ot w) = 4,7 (t) oi aioe (t), 
P(t + ) = df(t) + Ago¢ (t), 
P(E + w) = Ay, S(t) + Ao 9"(t), 
we derive, by putting t= 0, a,, =f(w), d,. = ¢’(w). The characteristic equation 
in this special case is 
“ (441 — 8) (Gag — 8) — Ayo My, = 0, 
whence A = a), + Ay. =f (w) + ¢’(w). 
If we now replace p(t) by Ap (t), we obtain the developments of the integrals 
J (t), ¢(t) in the form 
FO=LP A O+e:- PMAO+L---, 
b(t) = t+ Abit) + +++ + Maal) $y 
where the functions f, and ¢,, together with f, and ¢,, vanish fort = 0. Substi- 


tuting these developments in the two sides of the equation (118), after having 


replaced p by Xp, we find 
af 
"a =p (1) fn—1(t), 


whence we derive the recurrent relations 


t= ft f rPOhaOd, m= af rO ona, 


which enable us to calculate step by step all these functions by starting with 
Jo (t) =1, (4) =t. It follows that we may write 


d* bn 


dt2 = p(t) gr-i(d), 


+0 
(120) ‘Ae op > [fa (w) + 6, (w)]. 
n=1 


If the function p(t) is never negative, we see at once that all the functions 
In(t), en(t), ,(t) are positive fort >0. It follows that A> 2, and the equa- 
tion (119) has two real and positive roots, one greater and the other smaller 
than unity. The conclusion is much less evident in the other cases. If p(t) 
never takes on a positive value, it follows from a thorough study made by 
Liapunof * that the absolute value of A is less than 2, if the absolute value of 


® 
of pat 
0 


is less than or equal to 4. The equation (119) has in this case two conjugate 
imaginary roots, the absolute value of each of which is unity. 


*LiaApuNOF, Probleme général de la stabilité du mouvement (Annales de la 
Faculté des Sciences de Toulouse, 2d series, Vol. IX, p. 403). On the general theory 
of linear equations with periodic coefficients, in addition to the preceding paper, see 
also Floquet’s Annales de Ecole Normale supérieure, 1883, and Poincaré’s Les Méthodes 
nouvelles de la Méchanique céleste (Vol. I, chap. iv). 


152 LINEAR DIFFERENTIAL EQUATIONS (III, § 56 


IV. SYSTEMS OF LINEAR EQUATIONS 


56. General properties. Most of the theorems established for a 
linear equation can be extended without difficulty to systems of 
linear equations in several dependent variables. We shall assume 
in what follows, as we may without loss of generality, that these 
equations are of the first order (§ 22). Let y,, y,,---, y, be the n 
dependent functions, and « the independent variable. It follows 
from a general theorem (§$ 37) that the integrals have no other 
singular points than those of the coefficients. If we assign the 
initial values yf, y2,---, y? for a point « = #, which is not a singu- 
lar point, we can follow the analytic extension of these integrals 
along the whole of any path starting from a, and not passing through 
any of these singular points, which are known in advance. 

We shall suppose, only for simplification in writing, that we have 
a system of three equations with three dependent variables. Let us 
consider first the system of three homogeneous equations, | 


dy 
Ae yee a CaN, 


d 
(121) ae ty +e tou = 0, 


~ 


du 
Te + a + O% + eu = 0, 


where a, 6, c,--- are functions of the single variable x. If we know 
a particular system of integrals (y,, z,, w,), the functions (Cy,, Cz,, Cu,) 
also form a system of integrals for any value of the constant C. 
Similarly, if we know two particular systems of integrals, (y,, 2,, “,) 
and '(y,, #, u,), we can derive from them a new system of integrals 
depending upon two arbitrary constants, 


: CY, te CLYy C1, CE Cos C1u, ris Cy Uy 
Finally, if we know three particular systems of integrals, 


(Yy %y %)s (Yor %ay Uy)s (Yes Xr Uy)s 
the equations 
(122) 2= C2, + Cy, + Cye) 


‘ = CY, + Cy, + Ce¥Q 
U=Cyu,+ Chu, + 1Cyu, 


represent also a system of integrals, where C,, C,, C, are arbitrary con- 
stants. In order to assert that the expressions (122) represent the 
general integral of the system (121), we must make sure that we can 


III, § 56] SYSTEMS OF LINEAR EQUATIONS 153 


choose the constants C,, C,, C, in such a way that, for a given point 
x =a, not a singular point, y, z, w take on any preassigned values 
Yoo Xp) Uy Whatever. In order for this to be true, it is necessary and suf- 
ficient that the determinant of the nine functions y;, 2,, vu; (¢=1,2,3), 


Yen 
A=|¥%, *, U4), 
Y, &, Us 


shall not vanish identically. If this is true, we shall say that the set 
of three particular systems of integrals form a fundamental system. 


If A vanishes identically, the three particular systems of integrals reduce to 
two, or even to a single system. Suppose, first, that not all the first minors of A 
vanish simultaneously, for example, that the minor 6= y,z, — y,2, is not identi- 
cally zero. Let A be a region of the plane where 5 does not vanish. We shall 
determine two auxiliary functions K, and K,, analytic in the region A, such 
that we have 


(123) ¥3 = Kyy, + Ke%, @, = K,2,+ Ky2,, 


and since the determinant A is zero, these functions K, and K, also satisfy 
the relation 


(124) Us = Ku, + Kau. 


If we replace y, z, and wu in the first two equations of the system (121) by the 
preceding expressions for y;, Z,, Us, observing that (y,, 2,, U,) and (Y_g, 2, Ue) 
form two particular systems of integrals, we obtain, after simplification, the 
riers WiKi +y.K,=0, 24K, +2,K,=0, 

from which we derive Kj = K,=0. The functions K, and K, are therefore 
constants, and the relations (123) and (124) remain true in the whole region 
of existence of the functions y;, zi, wu. It follows that the system of integrals 
(Y¥35 Zg, Uz) is a combination of the other two. 

If all the first minors of A vanish identically, the three systems of integrals 
reduce to a single system. Since the elements of A cannot all vanish simul- 
taneously, let us suppose that y, is different from zero, and let us put y, = Ky,. 
From the relations y,2, — 2, Y, = 0, YU. — U,Y_ = 0 we derive also z, = Kz,, 
u, = Ku,. Replacing y, z, u in the first of the equations (121) by Ky,, Kz,, Ku, 
respectively, there remains y,K’ =0. Hence K is constant, and the system 
(Yo %y5 Uy) differs from the system (y,, Z,, u,) only by a constant factor. Similarly, 
the third system of integrals is identical with the first. It should be observed 
that ¥,, ¥,, Yg are not necessarily linearly independent ; for example, one or 
two of these functions may be zero, but not all three may be zero. 

The value of the determinant A may be calculated as follows. The derivative A’ 
is the sum of the three determinants 


7 te / 

Yr % Wy YW % WY Vigo yal 

7 a - 
N=l¥Yq 2% Ugltl¥e 2% Us|tly2 2 Uel- 


/ 
Y3g 2, Ug Yg 2% Ug Yg 23 Ug 


154 LINEAR DIFFERENTIAL EQUATIONS [III, § 56. 


Replacing the derivatives y;, z;, u; by their values obtained from the equations 
(121), these three determinants reduce, respectively, to — aA, —6,A, —c,A. 
We have, therefore, the relation A’ =— (a + b, + ¢,)A, and conneteente 


=x 
(125) A(x) = A(z) eg @tP tee 


When we know the general integral of the homogeneous system 
(121), we can deduce from it by quadratures the general solution of 
the non-homogeneous system 


l 
a + ay + bz + cu = f,(@), 
| dz 
(126) + a,y+bz+¢u= f,(x), 
d 
= + a,y + 6,2 + cu= f,(x). 
Indeed, if we make the change of variables defined by the equations 


(122), C,, C,, C, being considered as new dependent variables, the: 
system (126) is replaced by the pai system, 


4 + +4, =f(2), 
(127) z, “a + 2, = f(x), 
u, a + U, wa + u = f,(«), 
which is integrable by quadratures, for we derive from it 
= X.. (in ees 


Let us also observe that this transformation is unnecessary whenever 
we can determine directly a particular system of integrals (Y, Z, U) 
of the equations (126). In order to obtain the general integral 
of these equations, we need only add Y, Z, U, respectively, to the 
right-hand sides of the equations (122) which represent the general 
integral of the homogeneous system (121).* 


* A method analogous to that of Cauchy (§ 39) may also be employed. Let 
y= i (x, a), z= Yi (2, Q), U=Ti(@, a) (i=1, 2, 3) 


be three systems of integrals of the homogeneous equations (121), satisfying, respec- 
tively, the initial condition 


$4 (a, Q)= 1, Y1(Q, a)=0, 11 (a, a)=0, 
do (a, a= 0, Ye (a, @) =1, TT, (@, Qa)= 0, 
3 (a, a) =0, Vs (a, a)=0, 73 (a, a)=1. 


IIT, § 56] SYSTEMS OF LINEAR EQUATIONS afi) 


When we know one or two particular systems of integrals of the 
equations (121), we can lower the order of the system by one or two 
units. Suppose, first, that we know a single system of integrals 
(Y» %, U,), Where the function y, is not zero. The change of 
dependent variables 


y =Y,%, 2=2,Y+Z, uU=u,Y+U 


leads to a linear system of the same form which must have the par- ° 
ticular system of integrals Y=1, Z=0, U=0O. Therefore the 
coefficients of Y in these new equations must be zero. In fact, the 
transformed system is 


ad Y 
Y, 7, + 64 +eU = 0, 
az ad Y 
(128) vitae th Cie a 1, UO; 


dU adY 
de tt Gg 1 nF $Y = 0. 


If we replace dY/dx in the last two equations by its value derived 
from the first, we obtain a system of two linear homogeneous equa- 
tions in the two dependent variables Z and U. After integrating 
this system Y can be obtained by a quadrature. 

Suppose now that we know two independent systems of integrals, 
(Yy> 2 U1)» (Yo, %q) Ua)» Since the three determinants 


Y1%2 — YoFy Yy Ug — Yay @U, — #4, 


do not vanish simultaneously, as we have shown above, let us sup- 
pose that y,z, — y,%, 18 different from zero. The transformation 


Y= YY yes owe + 2,2; uU=UY+u,Z +0, 


where Y, Z, U are the new dependent variables, leads to a linear 
system of the same form having the two particular systems of 


It is easy to see that the functions 
Y= [Liilee) bx, &) +F2 (@) ba (2, @) +fs (2) ba (2, @)] day, 
= f “LFi(ee) Ya es ) +Fa(Q) Val, @) +fs (a) Vs (@, a) day 
Um [Lia ee) me, 0) + fae) 2 (@ 2) + face) a (2, @)] de 


form a system of integrals of the non-homogeneous equations (126). 


156 LINEAR DIFFERENTIAL EQUATIONS [III, § 56 


integrals’ (Y, =1, 7, =U,=0), (¥,=0, Z7,=1, U,=0).. The coeii- 
cients of Y and Z in the equations of the new system must therefore 
be zero, and this new system has the form 


adZ dU 
—-+AU=), ant i Ge 1 4eU = 9, 


as is easily verified. It is clear that this system is integrable by 
quadratures, since the last equation contains only U. 

The preceding methods may be extended to systems of » linear 
equations with n dependent variables. In order to obtain the general 
integral of such a homogeneous system, it is sufficient to know n 
particular systems of integrals which form a fundamental system. 
If we know p independent systems of integrals (p <n), the integra- 
tion reduces to that of a system of the same form with n— p 
dependent variables and to a number of quadratures. Finally, the 
general integral of a non-homogeneous system can be obtained by 
quadratures if we know the general integral of the corresponding 
homogeneous system. 


57. Adjoint systems. Given a linear homogeneous system with n dependent 
variables, . 


dy; 
(129) = Mi1Y1 +oee* + Mikye +++ + AinYn, (2, dot 2,e+, n) 
the linear system 
adyY; 
(180) ie =— Ay; Ya — +++ — Ae Yu — +++ — ni Yn, 


which is obtained from the first by replacing y; by Y;, and by changing the 
rows into columns in the determinant of the coefficients a;z, after having changed 
the sign of each element, is called the adjoint of the first. It is evident from the 
definition itself that this relation is a reciprocal one between the two systems. 
The integration of one of the systems (129), (180) involves that of the other. In 
fact, let (Y¥1, Yo, °**, Yn) and (Y,, Y,,--+, Yn) be any two particular systems of 
integrals of the two adjoint systems. From the relations (129) and (130) we have 


d 
ag (1141 t Yo¥e + soe Yun) =>) Vilas + set UKYe + e+ + AinYn) 


+> ui(- a4; Yy— +++ — Agi Ye— +++ — Ani Yn). 
% 


If we permute the indices 7 and & in the second sum, we see immediately that 
the coefficient of Y;y, on the right-hand side is 


Qik — Gz = 0, 


and the right-hand side is identically zero. We have therefore the relation 
between these two particular systems of integrals 


(151) Yi4i + Vote to°° + Ynyn = G, 


IM, § 58] SYSTEMS OF LINEAR EQUATIONS Tar 


where C denotes a constant. The knowledge of a particular system of integrals 
(Y,, Yo, +++; Yn) of the equations (180) furnishes therefore a first integral of 
the system (129), which is linear with respect to the dependent variables y,, y,, 

+, Yn. If we know the general integral of the adjoint system (180), the gen- 
eral integral of the given system (129) is represented by n relations of the form 
(131), where we take successively, for Y,, Y,,-++, Yn, a set of n independent 
systems of integrals of the equations (130). 

Particular attention has been paid to linear systems which are identical 
with their adjoint. In order to have this case, it is necessary and. sufficient 
that the determinant of the ay, be a skew symmetric determinant ; that is, that 
we have az + a,j = 0, whatever may be 7 and k, and consequently a; = 0. If 
(Yas Yor **s Yn) and (21, 2,,++-*, Zn) are two particular systems of integrals, the 
relation (131) becomes 


Yr + Yo %, + 2 eS, te Un&n = const. ; 
and if the two systems are identical, we have also 
yi tyet-->+y? = const. 


The integration of a linear system of the third order identical with its 
adjoint leads to the integration of a Riccati equation (§ 31, Ex. 2). The inte- 
gration of a system of the fourth order of that kind leads to the integration of 
two Riccati equations (see Ex. 15, p. 170). 


58. Linear systems with constant coefficients. If all the coefficients 
a, b, c,-+-+ of the equations 


d 
a tay + bz +cu= 0, 
dz 

(132) | pac a yet te 0, 
d 
5. + aay + bye + 0,4 = 0 


are constants, the general integral can be found by the solution of 
an algebraic equation. For let us try to satisfy these equations by 
taking for y, z, w expressions of the form 


(133) Ue, eas De", Uae. 


where a, B, y, 7 are unknown parameters. Substituting these func- 
tions for y, z, uw in the left-hand sides of the equations (132), and 
suppressing the common factor e’, we find the conditions 


(a+r)a+ bB + cy = 9, 


(134) aa+(b,+r)B+ey7=9, 
eee aag+bp+(c,+r)y = 9, 


158 LINEAR DIFFERENTIAL EQUATIONS (Ili, § 58 


which must be satisfied by values of a, 8, y which do not all vanish. 
For this it is necessary and sufficient that r shall be a root of the 
equation of the third degree, 


a+r b ee, 
(135) F(r)=| a “b,--+r c, |= 90, 
a, b, CUE: 


which is called the auxiliary equation. Having taken for r a root of 
this equation, the relations (134) are consistent and we can deduce 
from them values for a, B, y, at least one of which is not zero. 
To every root of the equation F(r)=0 corresponds therefore a 
particular system of integrals of the form (1383); there may even be 
several, as we shall see presently. If the auxiliary equation has 
three distinct roots 7,, 7,, 7, each one furnishes a particular system 
of integrals. These three systems are independent, for, if they were 
not, we could express e’s* as a linear combination with constant 
coefficients of e:* and of e”*, which would be absurd. We can there- 
fore, in this case, obtain the general integral of the system (132) 
after we have solved the equation F(r) = 0. 

It remains to treat the case in which the auxiliary equation has a 
multiple root. Let us denote by f(r), ¢(7), y(7) the three cofac- 
tors of the auxiliary determinant corresponding to the elements of 
the same row, for example, the first. The last two equations of 
the system (134) are always satisfied for any value of r by taking 
for a, B, y quantities proportional to these cofactors; if r is a root 
of F(r) = 0,.these values of a, B, y also satisfy the first of the equa- 
tions (134). It follows from this that if r is a root of F(r)=0, 
the functions 


Y= Fhe, z= (re, u = p(rje* 
form a particular system of integrals.. Now let us suppose first that 
the equation F(7r) = 0 has two roots, r, and r,, whose difference is 


very small. Each of them furnishes a system of integrals, and the 
functions 


FoyeFofeye™ gpreP— diet prae* — ore 


votw 4 LO me ‘ I a 


are also integrals. If we now let 7, approach 7, and pass to the limit, 
we may conclude that if vr, is a double root of F(7”) = 0, the two 
groups of functions, 


(1) fy Ji (7,) en", od a p (7,) e", u= Y (7,) en", 


III, § 58] SYSTEMS OF LINEAR EQUATIONS 159 


0 0 
i an L7(r) ee rar, Lf pany a L¢ (7) Ol pee r,3 
ay) 
Corr or Ly) ener) 


form two systems of integrals. Similarly (see § 44), if the equation 
F(r) = 0 has a triple root r,, we can add to the preceding two groups 
the group of three functions, 


e fe 
=O Ky =a ldMe™bery 
arn er ar 


g? 
bef Or? Ly (7) har 


which form a third system of integrals. 

Let us now consider first the case where the equation F(r) = 0 
has a double root 7, and a simple root 7,. If the double root 7, does 
not cause all the first minors of the auxiliary determinant to vanish, 
we may suppose that at least one of the cofactors f(7r,), 6(7,), ¥(,) 
is not zero, for we can evidently replace, in the reasoning which pre- 
cedes, the first row by the second or the third. Suppose, for example, 
(7) # 9. The two systems of integrals (1) and (II) are independ- 
ent, for y, is equal to the product of e” and a binomial of the first 
degree af(r,) + /'(r,). As for the simple root r,, it furnishes a third 
system of integrals which, for the same reason as above, is not a 
linear combination of the first two. 

The reasoning fails if the double root r, makes all the first minors 
vanish, for the system (I) reduces to the trivial solution 


Y,=2,=4,=0. 


But in this case the three equations (134) reduce to a single equa- 
tion when we replace in it r by 7,. If, for example, ¢ is not zero, 
they reduce to the single equation (a + 7,)a@ + 0B + cy = 0, and we 
can take the two constants @ and £ arbitrarily. If we take, first, 
(a=1, B=0), then (@=0, B=1), we obtain two independent 
systems of integrals of the form (133). A double root of F(r)= 0, 
therefore, always furnishes two particular independent systems of 
integrals. 

Suppose, finally, that F(r)= 0 has the triple root r=7r,. If this 
root r, does not cause all the first minors of the determinant to 
vanish, we may suppose, for example, that f(r,) is not zero. The 
three particular systems of integrals (1), (II), (III) are independent, 


160 LINEAR DIFFERENTIAL EQUATIONS [III, § 58 


for the coefficients of e” in y,, y,, y, are respectively of degrees 
O. 2) 2a. 

If the triple root 7, causes all the first minors of the determinant 
to vanish, we can determine first of all two independent systems of 
integrals of the form (133), as we have just explained in regard to 
the case of the double root, and we can then obtain the general 
integral if we can find a third system independent of these two. 
Developing the expressions in (III), and noting that 

12 (7,) =¢ (7) ian v(r)= 0, 


we find 
y, =e [2a f(r) + P'~)I) a, =e" [2ad(r,) +o") ], 
u, = e1* [(2ay'(r,) +0") 1 
and this system of integrals is certainly independent of the first 
two unless we have at the same time /f'(7,) = ¢'(7,) = w'(7,) = 0. 
Hence we obtain in this way a new system of integrals, unless the 


triple root 7, also causes the derivatives of all the first minors to 
vanish. Now this cannot happen, as we see at once, unless we have 


b=c=4a,=¢,=4,=—6,=0, a=b=c=—T7, 


and the system (132) reduces to three identical equations, 


In this case, which may be considered as a limiting case, the three 
equations (134) are satisfied identically, when we replace r by 7, in 
the expressions of (133), for any values whatever of the parameters 
a, B, y. Summing up, to a triple root of the auxiliary equation there 
always correspond three particular independent systems of integrals. 


Generalization. Similarly, a system of n linear equations with constant 
coefficients 4 
dy, ay; 
ne + A141 oP 13242 ad a A1nYn = VY, 
(136) 


*9 


d 
a + On Y1 + On2Y2 + ++ + AnnYn = 9, 
may be integrated by finding particular systems of integrals of the form 
(137) Y, = a,e*, Yo = a, 6, see, Vag eae, 


where @, @,+*+, @n, 7 are unknown constants whose values are to be deter- 
mined, We are thus led to n equations of condition 


III, § 59] SYSTEMS OF LINEAR EQUATIONS 161 


(44 + Yr) ay + Ayy X fies + Gin. 0, 
(138) Ay, @ + (Ang + 1)Ay + +++ +danAn = 0, 


See OR ens hic oe hn, eee ae 
Ani &1 + Ane He + eee + (Qnn + r) aX, = 0, 


which give for the unknown quantity r the auxiliary equation 


Ayy+r A192 ey An 
(139) Ri eM, co ee | <0) 
Ant Ang se* Onan + 7 
If this equation has n distinct roots r,, 72, +++, %, we obtain by this method 


n particular systems of integrals of the form (137) and, consequently, the gen- 
eral integral. If there are multiple roots, the discussion is somewhat more 
complicated. Let r, be a p-fold root; to obtain from this root particular sys- 
tems of integrals of the equations (136), we may proceed in two ways. On the 
one hand, applying d’Alembert’s method, as in the case of three equations, we 
can obtain p systems of integrals corresponding to that root. These integrals 
will be independent only if r, does not make all the first minors vanish. On the 
other hand, if r, makes all the minors formed from n— gq +1 rows of the deter- 
minant vanish, without making all those of n — g rows zero, that root furnishes 
q systems of integrals of the form (187), for the n equations (188) reduce to 
n—q independent equations when we replace r by r,. Combining these two 
methods, we find that they always furnish p independent systems of integrals. 

Practically we can obtain all these systems by equating coefficients. In fact, 
by the combination just mentioned we should obtain a system of integrals 
depending upon p arbitrary constants, which is of the form 


yy = eit F(z), Yo = ei" P(x), e885 Yn = ei P, (x), 


where P,, P,,---, P, are polynomials of degree » — 1 or of lower degree. If 
we leave the coefficients of these polynomials as unknown, and if we substitute 
in the given equations, we shall obtain a certain number of relations between 
these coefficients, which enable us to express all of them in terms of p of them, 
which may be taken as arbitrary constants. 


59. Reduction to a canonical form. Every linear system with constant coeffi- 
cients can be reqgoed to a simple canonical form the integration of which is 
immediate. <¢ 

Let us write this system under a slightly different form, 


Yi = Uy + yoYg tere + UnYn, 
(140) Yo = Aq Yy + Aq Yq + +++ donYny 
Yn = An Yy + Ane Yq + +++ + AnnYny 
where y; denotes dy;/dz. If we take n dependent variables, Y,, Y,,---, Yn, 
linear in terms of y;, Yo, +++ Yns 


(141) Yi = bY, + +++ + OinYn, (a=iy 2,+++, 7) 


162 LINEAR DIFFERENTIAL EQUATIONS [III, § 59 


where the coefficients bj are constants whose determinant is different from 
zero, the system (140) is replaced by a system of the same form, 


Yy = Ay Yy + Aye ¥o + +++ + AinYan, 
(142) . Yo = Ag Y, + Ag ¥2 + +++ + Azn¥n, 


Ne == AmY, ae AnY, meee AnnYn, 
obtained by replacing the variables y,, ¥,, +++, Ym, in the expressions for Y;; 

Vase bi yj gee Le 5 Din Yn = bi (444 Yy fs 22 st in Yn) sw 

Se Din(Gn1 V1 eS Oke is Onn Yn), 

by their values given by the equations (141). If we consider the equations (140) 
as a linear substitution carried out on the variables y,, Y2,+++, Yn, and Y,, Y,, 
--+, Y, as n new variables, the preceding calculations are precisely those which 
we must make in order to find the new linear substitution on the variables 
Y,, Yo,:++, Yn, Which corresponds to the linear substitution (140). Now we 
have seen that by suitably choosing the variables Y; (§ 48) we can reduce every 
linear substitution to a simple canonical form.* In this canonical form the 
variables separate into a certain number of distinct groups, such that the 
substitution which the p variables Y,, Y,,---, Yp of the same group undergo 
is of the form . 


(148) pe a%y, | Yess (at Ya). +2, lp a8 Wp te 


We can therefore, by a suitable change of variables of the form (141), always 
reduce the integration of the system (140) to the integration of\a certain num- 
ber of systems of the form (143), where Y; = dY;/dz. 

The integration of this system is immediate, but it is preferable to employ a 
somewhat different canonical form. For this purpose let us set Y; = s'z;(s 4 0). 
The system (143) becomes 


dz dz 
(144) rect. 7 eth vee, Te ee + %-1- 


This new canonical form is unchanged if we multiply all the dependent vari- 
ables by a factor eA”, except for the change of s to s +d; and it is applicable 
also to the case where the auxiliary equation has zero for a root. 
The general integral of the system (144) is represented by the equations 
zje— se = G, PE + pomtT a 
(t—1)! (ti — 2)! 


or by equivalent equations obtained by solving for the constants C; 


+ +--+ Oj-12+4+ C; (t= 1, 2, »++, p) 


2 
(145) ee Coe (% — ele ot = Gey (« — 2%. + 5%) e— st = O,, 


int 
aa canes (1=1, 2, +++, p) 


i a ee ee a Se ee ere 

* We supposed before that the determinant of the substitution was not zero, 
whereas the determinant formed by the coefficients ai may be zero. But if we 
change yi to eA” z;, the coefficients a41, dg2,°+*, dnn are diminished by A, while the 
aik’8, Where i 4k, do not change. We can therefore always choose \ in such a way 
that the determinant of the new coefficients shall not be zero. 


x p 
{#2214 SS es coe (— 1)t3 


III, § 60] SYSTEMS OF LINEAR EQUATIONS 163 


60. Jacobi’s equation. Let us consider again a system of three 
linear equations with constant coefficients, which we shall write in 
the form 


dx 
—=ar+ by+ cz, 


dt 
dy 
(146) ae + by + ¢,2, 
dz 
ite a,% + by + ¢,%, 


where ¢ denotes the independent variable. Let us add these three 
equations, after having multiplied them respectively by ydz — zdy, 
zda — xdz, xdy — ydx. The relation obtained is 


(ax + by + cz) (ydz — zdy) + (a,x + b,y + ¢,2) (eda — adz) 
+ (a,x + by + ¢,2) (ady — ydx) = 0 
and it is homogeneous in a, y, z. Hence it can be replaced by a rela- 


tion between «/z and y/z. Indeed, if we put « = Xz, y = Yz, and 
divide by 2°, this relation takes the form 


(147) 


—(aX + 6¥+c)dY+(a,X +6,¥ + .¢,)dx 


(148) +(a,X + b,¥ + ¢,)(Xd¥ — YdX)=0, 


which is exactly Jacobi’s equation (pp. 11 and 32). 

Let x= f(t), y= ¢(t), 2 =Y() be a system of integrals of the 
equations (146). As ¢ varies, the point whose homogeneous coérdi- 
nates are x, y, # (and whose Cartesian coérdinates are X = x/z, 
Y = y/z) describes a plane curve © which is, by the preceding argu- 
ment, an integral curve of Jacobi’s equation (148). The integration 
of Jacobi’s equation therefore reduces to the integration of the sys- 
tem (146), that is, to the solution of an algebraic equation of the 
third degree, as we have already seen. 


If the auxiliary equation has three distinct roots s,, 8,, 8,, the general inte- 
gral of the system (146) is, according to the preceding paragraph, of the form 


(1) Pert (Ny np Oe tm Ua) Heats Cs, 


where P, Q, R are three linear homogeneous functions of z, y, z. It is easy to 
derive from these equations a homogeneous combination of degree zero which 
does not contain the variable ¢, 


(a) P83 — 85 Q1 — 8, R8.— 8, — Ee 
which is the same result that we obtained before by another method. 


The case in which the auxiliary equation has a double or a triple root 
can also be easily treated, The equations representing the general integral form 


164 LINEAR DIFFERENTIAL EQUATIONS [ III, § 60 


either two groups or a single group. In the first case these equations are of 
the form 


(IT) Bem Oy, (Q—tP)e-4s* = C,, Re-s' =, Ge, 
and in the second case, of the form 
2 
(III) Pe-s'=C,, (Q—tP)e- C2, (n- 1945) eit = Cs, 


where P, Q, R denote in each case three linear homogeneous functions of 
x,y, z. From (II) we derive the following homogeneous combination of zero 
degree, independent of t: 


Pear! 
(8) zen PPLE, 
and from (III) the combination 
2PR— Q? 
(7) ot pre, oe 


The relations (a), (8), (vy) represent the three forms possible for the general 
integral of Jacobi’s equation. 


61. Systems with periodic coefficients. Let us consider first, for simplicity, a 
system of three equations of the form (146), whose coefficients a, b, c,--- are 
continuous functions of the variable ¢; each of which has the period w>0. 

Let (£1, ¥15 21) (Les Yor 22). (Xs, Yg, %g) be three independent systems of inte- 
grals. Then the functions 


A; (t) = a(t + w), Y;(t) = yi (é + @), Z(t) = z(t + w) 
also form a system of integrals, and we have consequently three groups of. 


relations of the form (§ 56), 


Xj = A171 + Aig Te + AisZs, 
(149) Yi = Gi1Y1 + Gzye + Gs Ys, => To) 
Zi = Ay 21 + AigZ% + AisZs, 


where the a;z’s are constant coefiicients whose determinant H is not zero. We 
have, in fact, the relation 


Ap tee oh S| 
BE NGS OOF 5 NY a) ean eae 
X, Ys, Z, v3 Y3 %% 


or, by (125), reasoning as we have done several times (§§ 38, 56), we may write 
the value of H in the form 


(150) jee dN Coen e os 
If the variable ¢ is increased by the period w, the three functions 


x, (t), X(t), Lz (t) 
undergo a linear transformation whose determinant is different from zero, 
defined by the relations 
AY = G12 + Ayo Xo + Ay3%5, 
(151) Ag = Mg + Age Ly + Migs Le, 
Ae gy Xy + Ogg ty + Ogg Xp, 


Ill, § 62] SYSTEMS OF LINEAR EQUATIONS 165 


and the two other systems of functions, (¥,, Yo, Y¥s)5 (21) 22) 2g), undergo the same 
transformation. Now we know that it is possible to replace the three functions 
£1, XL, L, by three independent linear combinations with constant coefficients 
such that the equations (151), which define the new linear transformation, take 
on a simple canonical form. Taking the same linear combinations of the func- 
tions (¥,, Yo, ¥3) and of (z,, 2, %,), We obtain three systems of functions which 
are transformed by the same linear substitution of canonical form when t is 
changed to ¢t + w. 

The reasoning is evidently general and applies to every linear homogeneous 
system in n dependent variables with periodic coefficients. Let y,, %,-++; Yn 
be these n dependent variables. We can determine n independent systems of 
integrals (yi, Yai, +++, Yni) (¢ = 1, 2, +++, n) such that the n functions 


Ykly Yk25 Gr Ykn 
undergo a linear substitution of canonical form when ¢ changes to ¢ + w, this 
linear substitution being the same for all the indices k. The consequences are 
the same as those which have been developed above (§ 54). All the integrals 
are expressible as the product of an exponential factor of the form e** and 
another factor which is either a periodic function of ¢ or a polynomial in ¢ 
whose coefficients are continuous periodic functions of t. Let 


Yel C Pyy i ee ye So hogy, anita! of Yale Sn 
be a particular system of integrals, where z,, 2,,+++, Zn are polynomials in ¢, 
with periodic coefficients, of which at least one is of degree p — 1, and of which 


none is of a degree greater than p—1. From this system of integrals we can 
derive (p — 1) other systems of the form 
Vig = ett Dz, Yoo = ext Dz... = 12°’ Yo €! Den, 
was, h ws, oes, 

Vine 6 De as Ue peaest DP XZ, ee, Yan = Cra ee, 
where the derivatives Diz, are taken regarding the periodic coefficients of the 
powers of t as constants (§ 54). All the systems of integrals of the given equa- 
tions can thus be derived from a certain number of them. The actual formation 
of these integrals, of which we know only the analytic form, depends, above all, 
on the solution of an algebraic equation of the nth degree, which is called, as 
before, the characteristic equation of the system. The coefficients of this equa- 
tion can be obtained in general only by approximations, as in the case of a 
single differential equation of the nth order (§ 55). 


62. Reducible systems. Let us consider a system of linear homogeneous equa- 
tions of the form (140), whose coefficients are real, continuous, and bounded 
functions of the real variable ¢ for all the values of that variable greater than 
a certain bound ¢,, and let us suppose that we apply to this system a transfor- 
mation of the form 

(152) Zi = bi1y1 + DigYe + +++ + OinYny (i =1, 2,+++, ) 
where the coefficients bz, satisfy the following conditions: 

1) They are real, continuous, and bounded functions of the variable ¢ for 
t> to; 

2) They have derivatives satisfying the same condition ; 

3) The reciprocal of the determinant of the 6,’s is bounded. 


166 LINEAR DIFFERENTIAL EQUATIONS [IIl, § 62 


If we take the functions z; for new dependent variables, it is clear that the 
system (140) is replaced by a linear system of the same kind as the first. We 
have, in fact, vee ; 

a On Oa eos 
or, replacing yj, y3,-:-, y, by their values obtained from the equations (140), 


° = Ci Y1 + CigYo2 t+ ++ + CinYn; 
where the coefficients c;, have the same properties as the coefficients ay. We 
have now only to replace 71, Y2,-+-, Yn in these last equations by their expres- 
sions in terms of the new dependent variables 21, z2, +--+, Z, obtained from the 
equations (152). | 

If it is possible to choose the coefficients b,, of the transformation in such a 
way that the new system will be a system with constant coefficients, the system is 
said by Liapunof to be reducible. See page 242 of his paper cited in the footnote 
on page 151. 

Every system whose coefficients are real, continuous, and periodic functions, with 
the same period w, is reducible. 

In fact, let us consider the adjoint system, which is also a system with 
periodic coefficients. Let s be a root of the characteristic equation and @ the 
corresponding characteristic exponent. We shall suppose, in order to consider 
the most general case, that to this exponent @ corresponds a group of p par- 
ticular systems of integrals of the form previously considered. This group will 
therefore furnish (§ 57) p linear first integrals of the given system, which will 
be of the form 

em (2141 + Z2Y2 + +++ + ZnYn) = C1, 
ert (y; Dey + yo Dea + +++ + Yn Den) = Co, 
e%t (yy DP—121 + Yo DP 122 + +++ + YnDP—12n) = Op, 
where 21, Z2,°++, Zn are polynomials in ¢t, of degree p — 1 at most, with periodic 
coefficients, and where the derivatives Di are taken regarding these coefficients as 
constants. Arranging these first integrals with respect to ¢, we may write them 
in the form 


eet Th eos AY he +¥,|=¢ 
(PET pena ines 2 ant 


tp—2 tp-8 
158 ce | Ce ry ce ey A eee 
ne) 7 ear e243) ica | ss 


where Yq, Yo,++-, Yp are independent linear combinations of yj, Y2, +++, Yn, With 
periodic coefficients. For if they were not independent, we could derive from 
the equations (153) a relation between the arbitrary constants Ci, Co,---, Cp 
and the variable t. If we take the linear combinations Yi, Yo,---, Yp for 
dependent variables, the relations (153) represent precisely the general integral 
of the linear system of equations (§ 59), 

aya dY2 dY, 


(154) Dries ae Ti a ea oe Sq Daas pe Pal 


Ill, Exs. | EXERCISES 167 


Proceeding similarly with all the groups of first integrals furnished by the groups 
of integrals of the adjoint system, we see that the given system is transformed 
into a linear system with constant coefficients by means of a transformation of 
the form 


(155) Yi = $11Y1 + Pi2Yo2 + +++ + PinYn, 


where the coefficients ¢;, are periodic functions with the period w. 

The reciprocal of the determinant D of the ¢j’s is bounded for t > to, for 
we shall show that D does not vanish for ¢ >t). Indeed, if we consider n inde- 
pendent systems of integrals (y1i,-+-, Yni) of the first system and the corre- 
sponding n systems (Yxi, ---, Yne) of the transformed system, the determinant D 
is equal to the quotient obtained by dividing the determinant of the Y,;’s by 
the determinant of the y;;’s, and we know that these last two have finite values 
different from zero for all finite values of t. It follows that the absolute value 
of D remains greater than a certain positive minimum for all values of ¢ 
between ¢, and t)+w. 

In order to complete the proof, we may suppose that the characteristic equa- 
tion of the adjoint system has no real negative roots; for, by § 54, any root is 
replaced by its square if we consider the period 2w instead of the period w. 
If the characteristic equation has only real positive roots, we may evidently 
suppose that all the functions ¢4 which appear in the equations (155) are real. 
Then that transformation actually satisfies all the required conditions. More- 
over, all the characteristic exponents are real, and the transformed system has 
real coefficients. But if the characteristic equation of the adjoint system has 
conjugate imaginary roots, to each group of p linear combinations, such as 
Y,, Yo,-::, Yp, in which appear imaginaries we can associate the group 
formed by the conjugate imaginaries. Hence, combining them in conjugate 
pairs, it is clear that we again obtain a system with real constant coefficients 
by means of a transformation of the desired form with real coefficients. 


EXERCISES 
1. Integrate the linear equations 


yOY) —2y” 4+ y= Ae + Be-~+ Csinz + Deosa, y+ y7%= 
Yr—Yy’ ty —y=2e— 4 cosa, 
y" —8y + 2y = (ax + b) et + ce- 27, 
2y’” — 9ay’ + 97 =14 224 82? Logz, 
vy” —2ey’ + 2y=—224+ prtg, 
x8y/” — 8 x2y” + T2y’ nat sae 

dx 

2 Vi¢at | 

ry’ — 9a?y” + 387 ay’ — 64y = at [a + b Loge + c(Log2z)*], 
x2y”’ + 2ry’ —2Qy=xcosx— sing, 
xy” + 3ay’ + y =f (2). 


If f(x) is analytic in the neighborhood of the origin, prove that this last equa~- 
tion has a particular integral analytic in the same neighborhood. ¢] 


xy” —32y’ +4y = 272+ 


168 LINEAR DIFFERENTIAL EQUATIONS [IIl, Exs. 


2. Integrate the systems of:linear equations 


dy *-az adz7 an dx dy 
— — — =) i =0 Sa sgh ecaetinle nema ha 
(2) TN ERE LCi oo cena. 
a2 x a 


py May est eee 


dx? dx * 
(Y) Be 
wi iteae 3S Qy+3z2=-6-%; 
oe ee Be 2yt+ 
dx dy is 
5 — —z=0, —-—2z=0, —+2r—2=0; 
o aie at ot 
dx dy a 
—+2r—y=0, — —4z=0, —+4z-—27=0; 
SA Resterines yan ene) at 


dx dx 


Lee eae 0. 
dx 


YAN CE LY a eat 2(1— Ayu =), 
(n) Z+ryt24+2(A—1lju=), 
w+ rAy + (2A—1)u=0. 
3. Find the general integral of the equation 
(22% + 1)y” + (4a — 2)y’ — 8y = (627 + &— 8) 


from the fact that the homogeneous equation has a particular integral of the 
form e”*, where m is a constant. [ Licence, Caen, 1884. ] 


4. Prove that the differential equation 
(227 —1)y”=n(n+1)y, 


where n is a positive integer, has a polynomial P(x) for integral. From this 
prove that the same equation has a second integral 


r+1 


Pog (=+ i) + Q, 


where Q is also a polynomial. [Licence, Paris, 1890. ] 


5. The linear differential equation 
—(@+tutr)yt+puy =), 
where uw and » are two positive integers, has a polynomial y, = P(x) as an inte- 


gral. Hence prove that it has a second integral y, = e™Q(x), where Q(z) is also 
a polynomial. [Licence, Paris, 1903. ] 


6. Find the necessary and sufficient condition that the linear equation 
y”’ + py’ + qy =90 may have two independent integrals, y,, y,, which satisfy 
the relation y1y¥2=1. Assuming that p = — 1/2, find the coefficient qg and the 
general integral. [Licence, Paris, 1902.] 


7. Derive the formula (23), p. 111, from the formula (11), p. 106, which 
gives the value of the determinant Nap Ye glee SMa EY 


Ill, Exs.] EXERCISES 169 


8. Bessel’s equation, 
ry’ + 2(m+1)y + 2y =0, 


has as a particular integral the function represented by the definite integral 
1 
Yy =f (1 — 2?)™ cos xz dz, 
0 
provided that the real part of m is greater than — 1. If m is a positive integer, 
that integral is of the form (see Vol. I, end of Chap. V, Ex. 20, 2d ed.; Ex. 21, 
ey, 2.4.6---2m(Using + V cosa), 


where U and V are polynomials in 1/x whose coefficients are all integers, and 
the general integral is 
y = C(Usinz+ V cosz) + C’(V sing — U cosa). 


[ HERMITE. | 
9. The integration of the system of linear equations 


dy dz 
Pe ort se ae 101s 30; 


where a, b, a,, 6, are any functions of x, reduces, on putting y = tz, to the 
integration of Riccati’s equation, 
dt 
Fags b+(a—b,)t—a,?=0, 
and to the calculation of f(a + b,) dx (see ftn., p. 112). 
10. The ratio z of two independent integrals of the linear equation 
y+ py’ +qy=0 
satisfies the differential equation of the third order, 


gl’ 3 2/\2 1 , 
—-3(5) ie a Cater 


z 2 \2 
11. Given the differential equation 
(Z) a(y” —y’)— ay = 90, 


where a is constant, how must we choose the path of integration Z so that the 
function y (x) represented by the definite integral 


y (x) 24) er zer (2 eal dz 
(L) 


shall be a particular integral of (#)? Show that the equation (#) has a par- 
ticular integral, which can be expressed without any sign of quadrature, when 
a is an integer. Deduce from it the general integral, and express it in terms 
of the smallest possible number of transcendentals. 
[Licence, Paris, July, 1908.] 
12. Determine the two functions P(t) and Q(t) so that the function y 
represented by the expression 


y=(e— a) {FOP Od + @—v f FO Qa 


shall be an integral of the differential equation y” = f(z) for all possible forms 
of the function f(z). [Licence, Paris, October, 1907. ] 


170 LINEAR DIFFERENTIAL EQUATIONS [III, Exs. 


13. The general integral of the linear equation 
cy’ +[n+2P(x)]y¥ +2*+1Q(r)y=90 
where P(x) and Q(x) are analytic in the neighborhood of the origin, is single-_ 
valued in this neighborhood. The letter n denotes an integer greater than unity. 


14*. Every equation of the form 
ls 


mood + g—Pp— 1Q,() sath on 


+ 2Qn-p- (2) S SM aS (a) 8 +. “++ Qn(e)y = 9, 


“LP 


where Q,, Q,,+++,; Qn are analytic functions in the neighborhood of the origin, 
has an analytic integral in the same neighborhood ; and the value of the inte- 
gral, as well as the values of its first p — 1 derivatives, may be arbitrarily chosen 
for x = 0, provided the equation 


(r—p)---(r—n +1) +Q, (0) (7—d)--- 4% — n+ 2) + +--+ Q—p(0)= 
has no integral root greater than p — 1. 
[E. Goursat, Annales de l’ Ecole Normale, 1888, p. 265.] 
Note. By an artifice analogous to the one which was used in § 50, we are led 
to prove the proposition for an equation of the form 
dpu_ = M (eo 
dP Rn tat 
r 


du 
+ 4u), 


where we have put dn—py 


(Yh ry’ coe np 
yt ry + 1" dan=e 


15*. Let 2 be a system of four linear equations identical with its adjoint 
(p. 156) 


dy; ‘ 
(£) oe = M1 Y, + Ai2Vq + GisYs + U4 Vy. ((=1, 2,3,4) aaz+ an =0 


This system has the first integral y7 + y3 + y2+ y2= C. If we suppose C=0, 
the preceding relations are satisfied by putting 
Y, = p(n—6), Yo = p(1+ &y), Ys = pi(1—én), Y, = pi(n + &). 


Substituting these expressions for y,, ¥,, Ys, y, in the equations (Z), we obtain | 
the system of three equations 


= (dai + tds) (9 — &) + 2 tes + (sq + tdos) (0 + £), 

= (2 + 43) (1 + 9?) + i (Gis + eo) (1 — 7?) + 24 (go + Qy4) 0, 

= (a1 + M43) (1 + &) + 2(Qa1 + Gog) (1 — £7) + 2% (32 + G41) &, 
of which the last two are Riccati equations, Let » =f(z, C), = ¢(z, C,) be 
the general integrals of these two equations; then the general integral of the 
equation in p is given by the equation 

yitle Pile 
BODO. (ivi 
[E. Goursat, Comptes rendus, Vol. CVI, p. 187, and Vol. CXLVIII, p. 612.] 


ll, Exs.] EXERCISES 171 
16. Prove the relation 


fe @ae fo eae ff 'o @ae fs@)ae 
= ain f le (x) — o(y)]"—1F(y) dy, 


in which the left-hand side contains n integral signs, by proving that the two 
sides are particular integrals of a linear differential equation of the nth order 
satisfying the same initial conditions, 

17*. Prove that the integral ¢ (x, a) of the linear equation F(y) = 0 (p. 108), 
considered as a function of the variable aq, is an integral of the adjoint equation 
G (z) = 0, after having replaced x by a. 

Note. It is seen that the integral of the equation F(y) = 0 which, with its 
first (n — 1) derivatives, takes on the same values for ¢ = Ly as a function 7 (zx) 
and its first (n — 1) derivatives, has the form 


y=n(2)— f{ F@dse, a)da, 


where z = (a). The integral on the right must depend only upon 7 (2), 7 (2p), 
T’ (Xo), +++, 7-1) (x)). Now we can also write (§ 42) 


[FO o@, ada = (¥[z, 6, a)}27" — [26 [o(@, a)]da, 


and it is clear that the preceding condition is not satisfied unless we have 
G[¢(e, a)]=90. It follows readily that the functions ¢;(x) defined by the 
equations (A) (ftn., p. 109) form a fundamental system of integrals of the 
adjoint equation. 


CHAPTER IV 
NON-LINEAR DIFFERENTIAL EQUATIONS 
I. EXCEPTIONAL INITIAL VALUES 


The proof which has been given for the existence of integrals 
that take on given initial values really supposes that the right-hand 
sides of the given system of equations are analytic in the neighbor- 
hood of these initial values (§ 22). Restricting ourselves to the case 
of a single equation, we shall examine some simple cases in which 
that condition is not satisfied. 


63. The case where the derivative becomes infinite. Let tis consider 
an equation of the first order, 


dy 
(1) idan erd)s 
where the right-hand side f(x, y) becomes infinite for the pair of 
values x =2,, y= y, in such a way that its reciprocal 


i 


S(@ Y) = aN 


is analytic in the neighborhood of this pair of values. We can write 
the preceding equation in the form 


: AL ae aay 
@ EF 22 


regarding y as the independent variable and «x as the dependent 
variable. But since the right-hand side f(z, v) is analytic by hypothe- — 
sis fora =2,, y = y,, Cauchy’s theorem applies to the equation (2). 
Hence there exists an integral, and only one, which approaches 2, 
as y approaches y,, and that integral is analytic in the neighborhood 
of the point y,. The development of « — a, in a power series accord- 
_ ing to powers of y — y, necessarily commences with a term which 
is at least of the second degree, since dx/dy or f,(#, y) is zero for 
c=2,, y¥=y,, for otherwise f(a, y) itself would be analytic. Let 
this development of x — a, be 
(3) %— x =An(y—Y)™tAnsilYY¥—Y%)™t> +>. 

(m = 2, A,, # 0) 

172 


IV, § 64] EXCEPTIONAL INITIAL VALUES ib: 


From the equation (3) we derive a development for y — y, according 
to powers of (@ — x,)'” (see II, Part I, § 99), 
1 2 
(4) Y— Y, = 4,(@ — &)"+ a(a —a,)™+++>. (a, # 0) 
It follows that the equation (1) itself has this one and only this one 
integral of the form (4) which approaches y, as x approaches x,, and 
the point x, is an algebraic critical point for this integral.* 


64. Case where the derivative is indeterminate. The complete dis- 
cussion of all the cases in which the derivative becomes indeter- 
minate is much more complicated. Let us take first the equation 
studied by Briot and Bouquet,t 

(5) ay’ — by = Aye yg +. Ay ty as off" +:++= 6, Y)s 
where the right-hand side is analytic in the neighborhood of the 
point « = y = 0, and let us try to determine whether there exists an 


analytic integral which vanishes with x For this purpose let us 
substitute for y, in both sides of the equation (5), a power series 


(6) om -- oo + +++ + 6.2" + ++ 
After the substitution the coefficient of x” on the left-hand side is 
(n — b)c,, while the coefficient of x” on the right is a polynomial, 

J Gey Bs °° “> Vn} Cr yaeae 9 Ca t)} 
whose coefficients are all positive integers, and which contains only 
the coefficients c,,---,¢,_,, and some but not necessarily all of the 
coefficients a,, for whichi + h =n. We therefore obtain a recurrent 
relation for the coefficients of the series (6) : 
| (7) ‘Rs vai: b) c= P(a.,, Hoos sey Qons Cy Cy) Ter his'9 Cxay) 
(n = bh 2, ee -) 

This enables us to calculate these coefficients successively, provided 
that b is not equal to a positive integer. Let us first set aside this 
supposition. The relation (7) gives us 


2 
_ Ay Aggy + Ay 6, + B26} 
C= ) 6. = SS reey 


1 1-0 3 _ 2-6 


*In geometric language, we can also say that through the point (x9, yo) there 
passes one and only one integral curve, on which the point (Xo, yp) is an ordinary 
point, and the tangent at this point is the straight line x=. In stating the theorem 
we have tacitly assumed that the function /;(x, y) does not vanish for x= “9 for all 
values of y; for in this case the integral of the equation (2), which takes on the value 
xq for y= Yo, reduces to x= og, and the equation (1) has no integral which approaches 
Yo aS & approaches ‘rp. 

t Journal de ’ Ecole Polytechnique, Vol. XXI, 1856. 


174 NON-LINEAR DIFFERENTIAL EQUATIONS | [iv,§ 64 


and so on in this way. The value of the series (6) certainly repre- 
sents an integral of the equation (5) vanishing with x, provided that 
the series has a radius of convergence different from zero. In fact, 
the operations by which we have obtained the coefficients of this 
series are then valid (I, § 192, 2d ed.; § 186, 1st ed.). 

In order to prove the convergence of this series, let us observe 
first that if we give to n all the integral values 1, 2,---, to infinity, 
the fraction 1/(n — 6), which cannot become infinite, approaches 
zero. The absolute value of that fraction has therefore a maximum 
1/B, and we have for every value of the integer n, |1/(n — 6)| S1/B. 

On the other hand, let 


@(x2, Y)=A,2 + Ala? +A, 2Y +A Y* +e Aga Y* foe 
be a dominant function for $(a, y), having no constant term nor any 
term in Y. We might take, for example, a function of the form 


M ¥; 
®(x, Y) = —_—_—___- —- M—M-,; 


Ce ee 


but it is really not necessary to specify it completely in order to 
carry through the proof. The auxiliary equation 


(8) BY =a (3) 1g) 


has, by the general theorem on implicit functions (I, § 193, 2d ed.; 
§ 187, 1st ed.), an analytic root vanishing with x. Let 


(9) Y=Ca+Cya?+---+C,a"+--- 
be the development of this root in a power series. In order to cal- 


culate the coefficients C;, we can substitute this development for Y 
in the two sides of the relation (8). This gives the recurrent relation 


(10) BC, =P, (Ay AY Veabee AGH; Cl; C5; Teas Coa) 
where P,, denotes the polynomial which appears in the relation (7), 
in which a, has been replaced by A,, and c; by C;. 

But from the very way in which we have chosen the constant B 
and the function ®(a, Y), we have the inequalities 


rast 
ln —b| ~ B 


|ax/SAx, 
It follows that if we have _ 
le|< Cy [e,,< C,, ad Ceres Cal 


we have also |c,| < C,, since all the coefficients of the polynomial bo 
are positive integers, Now we have |a,,|< A,., and, consequently, 


IV, § 64] EXCEPTIONAL INITIAL VALUES 175 


|c,||<C,. Reasoning step by step, we conclude that the series (9) 
dominates the series (6). The latter is therefore convergent in the 
neighborhood of the origin. Summing up, if the coefficient b of y in 
the equation (5) is not equal to a positive integer, that equation has 
one and only one analytic integral that vanishes with x. 


In order to finish the study of the analytic integrals which vanish with «, 
we must still examine the case where b is equal to a positive integer. Suppose 
first b= 1; the first of the relations (7) reduces to a,,=0. If a,, is not zero, 
then there is no analytic integral fulfilling the condition. If a,, is zero, setting 
y = de, we are led to an equation, 


(11) NM = W (a, d) = digg + AyzX + Aggd? + -°*, 


where the function y (x, d) is analytic, provided that |z|<r, |A|< A, r and A 
being two suitably chosen positive numbers. Now this equation (11) has an 
infinite number of integrals which are analytic in the neighborhood of the 
origin, for we can choose arbitrarily the value \, of \ for « = 0, provided that 
we have |A,|< A. Hence in this case the equation (5) has an infinite number 
of analytic integrals vanishing with a. 

If b is equal to a positive integer greater than unity, the coefficient of x in 
the development of an analytic integral vanishing for 7 =0 must be equal 
to a,)/(1 — 6), and the transformation y = a,,x/(1 — b) + dz leads to an equation 
of the same form in which the coefficient of \ is equal to (1— b): 


an’ — (b—1)X= 4} % + Ay) X? + ayy ADH ees. 


By a succession of similar transformations we reach the case which has just 
been treated. Consequently, if the coefficient b is equal to a positive integer, 
the equation (5) has no analytic integral vanishing with «, or it has an infinite 
number of such integrals. 

Briot and Bouquet also investigated whether there existed non-analytic inte- 
grals approaching zero with z, and proved that the equation (5) has an infinite 
number of such integrals when the real part of b is positive. We can easily 
establish this theorem by means of the method of successive approximations. 
Let us first point out that if the real part of b is positive, we may, without 
lack of generality, suppose that the real part R(b)>1. In fact, if we make 
the change of variable z = x", where n is a positive integer, the equation (5) 
is replaced by an equation of the same form in which 0 is replaced by nb. 
We shall suppose, then, that we have R(b) >1, and that b is not an integer. 
As -we have just seen, the equation (5) has an analytic integral y,, which 
vanishes for z= 0. Hence, setting y=y,+u, the equation (5) becomes 


ru’ — bu= (a, y1 + u) — $ (2, 1) = uy (Z, U). 


Since the function ¢(x, y) does not contain any term of the form a constant 
times y, the function y (x, u) will not contain any constant term, and we can 
write the preceding equation in the form 


cu’ — bu = ular + But ---]. 


176 NON-LINEAR DIFFERENTIAL EQUATIONS __ [lv, § 64 


Let us now put w= dz, where \ denotes the new dependent variable. The 
equation takes the form 


(12) NV =Ala + prw-14 ..-J= F(A, 2, 2-3), 


where F denotes a power series with respect to the three variables \, x, 2°-1. 
In the plane of the variable z let us draw through the origin two rays whose 
inclinations are wy and w,(wy) < #, <, + 27), and let us consider the circular 
sector A limited by these two rays and an arc of a circle with the radius r 
described with the origin as center. If @ remains in the interior of A, and 
if |\| remains less than a positive number J, the function F(A, 2, 2?—1) will 
be analytic,* provided that the two numbers r and / are sufficiently small. 
Let us connect the origin with any point xz of the sector A by a straight line- 
segment, and suppose that we take for the initial value of \ an arbitrary value 
\, whose absolute value is less than 1. We can apply to the equation (12) the 
method of successive approximations (§ 29), which consists in taking the 
successive integrals 


x x 
PES +f F (Ay, £0 —1) dz, 2 = Ao +f F(x, 2, 2-1) dz, 
0 0 


and, in general, x 
hn = No +f F(\n—1, 2 ®—1) da, 
0 


all of these integrals being taken along the straight line. The fundamental 
hypotheses for the validity of the proof are satisfied. All the functions \;(z), 
he(x), +++ are analytic in the interior of the sector A, and the function d, (x) 
approaches a limit A(z) if the radius r has been taken sufficiently small. Hence 
the equation (12) has an integral which is analytic in the interior of the sector 
A and which approaches the value \, as x approaches zero. | It follows that the 
equation (5) has an infinite number of non-analytic integrals in the neighbor- 
hood of the origin, each of which approaches zero as the point « approaches 
the origin and depends upon an arbitrary parameter \,. This proves Briot and 
Bouquet’s theorem. 

The condition that the real part of b —1 be positive is essential. Indeed, if 
x approaches the origin, remaining in the sector A, its angle remains between wo 
and w,, and its absolute value approaches zero. Setting x = pe*’,b—1l=yu+ vi, 
we have 


(13) go —-1 = e(h+ vi) (logp+iw) — eh logp—vw ei(vlogp + Mw), 


As p approaches zero, w remaining included between the two limits wy) and w,, 
uw log p — vw becomes infinite in absolute value in remaining negative, and the 
absolute value of 2?-1 approaches zero. On the contrary, if the real part of 
b — 1 is negative, it is obvious that the absolute value of x®?—1 becomes infinite 
as © approaches zero, remaining in the sector A. The function F(A, z, 2-1) is 
not continuous at the origin, and the previous proof no longer applies. 
According to Briot and Bouquet, if the real part of b is negative, the equa- 
tion (5) has no other integral than the analytic integral vanishing for x = 0. 


* If x approaches the origin, remaining always in the sector A, the derivative of the 
function /' with respect to x may become infinite if the real part of b — 2 is negative, 
but that derivative does not appear in the method of successive approximations. 


IV, § 65] EXCEPTIONAL INITIAL VALUES ET 


But their proof, which is very similar to the one in the footnote on page 50, sup- 
poses that the variable x approaches the origin along a path of finite length that 
has a definite tangent at the origin, and this condition should appear in the state- 
ment of their theorem. In order to give some idea of the difficulty of the ques- 
tion, let us consider the function 2, supposing that the real part p of b is negative, 
while the coefficient v of 7 is different from zero. The absolute value of x? is equal 
to eMlogp—vw, Tf we make the variable x describe a curve that approaches the 
origin indefinitely, «log p does approach + o, but if we make the angle w in- 
crease in absolute value at the same time in such a way that the difference 
log p — yw remains negative and increases indefinitely in absolute value, the 
absolute value of x approaches zero at the same time as|z]. If »>0, we need 
only make the variable x describe the logarithmic spiral whose equation is 
p = ev#/2", for example; for we then have |x| = e—¥/2, and if the angle w 
approaches + o, |x|= p and |x| approach zero simultaneously. 

If the real part of b is negative and the real part of b/i is not zero, it follows 
from investigations of Picard and Poincaré that the equation (5) has an in- 
finite number of non-analytic integrals that depend upon an arbitrary constant 
and approach zero as the variable x describes a path such as the preceding, along 
which |x?| approaches zero. The contradiction between this result and the 
theorem of Briot and Bouquet is only apparent, since in the two cases entirely 
different conditions are assumed. In particular let us notice that if the variable 
x takes on only real values, it cannot turn an infinite number of times around 
the origin ; consequently there will be no other integral which approaches zero 
with x except the analytic integral, provided the real part of b is negative. 

The results of this discussion are easy to verify with the elementary equation 
ry’ = ax + by, whose general integral is y = az/(1— b)+ Ca? if b—1 is not 
zero, and y = ax Log2 + Gail db = 1. 


65. We shall limit ourselves to a few statements concerning the 
general case of an equation of the form 


Co Re ER ra 


where X and Y are power series which converge when 
Cae? ig) § Prey 
We are supposing, as we may without loss of generality, that it is 


for « = y = 0 that dy/dx becomes indeterminate. Setting y = vax in 
this equation, it becomes 


dv _ a+bv—v(a'+ b'v) + 2¢(, v) 
“de a' + blu + ay (a, v) ; 
where $(a, v) and y(a, v) are two power series which are convergent 
whenever |x| < rand |va~| <r. If the equation (14) has an analytic 
integral vanishing with a, the coefficient of x in the development of 
that integral is necessarily a root of the equation 


(16) a+ bv —v(a'+ b'v) = 0, 


(15) 


178 NON-LINEAR DIFFERENTIAL EQUATIONS __ [Iv,§ 65 


since the left-hand side of (15) is zero fora = 0. Let v, be a root of 
the equation (16). If we put v= v, + wu, the two functions 


p(x, v, + 2); Y (a, v, + u) 
are still regular in the neighborhood of the values « = 0, u = 0, and 
the equation (15) reduces to an equation of a form already studied, 


(17) oot = Aut Bete, 
provided that v = v, does not make a' + b'v vanish. Since the equa- 
tion (16) is in general of the second degree, we see that we can 
reduce the equation (14) to the form (5) in two different ways. 
Hence there are in general two analytic integrals and only two van- 
ishing for = 0. But these conclusions are applicable only under 
the most general conditions, where the coefficients a, }, a’, b' do not 
satisfy any special relation. 

The general investigation of the integrals, analytic or not, of the 
equation (14), which approach zero when a approaches zero (X and 
Y being two regular functions which vanish for « = y = 0), has been 
the object, since the work of Briot and Bouquet, of a large number 
of investigations. Although it has been possible to treat more and 
more general cases, the question is still not exhausted. Let us notice 
in particular just one remarkable circumstance which we have not 
yet mentioned. Let us take the equation 


d 
(18) at —* — by = an, 


and let us try to find, as above, an analytic integral of this equation 
which vanishes for «= 0. If we attempt to determine the coeffi- 
cients of the series (6) so that on substituting it in the equation (18) 
we arrive at an identity, we discover the relations 


a+ be, = 0, c, = be,, 2 ¢, = be,, sey NC, = OC, di, 2 
from which we derive 


a a Pigs nia 
Ci — 9 6, = Y ara 


i b ES Maun SMUD Mey nite” cane sbes Te 3 
We thus obtain a unique value for each coefficient, but the series 
which we obtain is divergent except for x = 0. The origin is an essen- 
tially singular point for ‘all the integrals, as is verified by direct 
integration. Similarly, the point «= 0 is an essentially singular 
point for all the integrals of the equation ay! + y* = 0; and all these 
integrals approach zero with ||. 


IV, § 65] EXCEPTIONAL INITIAL VALUES 179 


If we assign only real values to the variables x and y, and wish to construct the 
integral curves of the equation (14) (X and Y being, for example, two polyno- 
mials in e and y with real coefficients), it is very important to know the form of 
these integral curves in the neighborhood of a point common to the two curves 
X=0, Y=0. We shall study from this point of view the simple equation 


d ax +6 
(19) Ya e, 
de wxt+ by 
which can be integrated by elementary methods (§ 3) by the substitution y = tz. 
We can integrate it in a more symmetric way by replacing the equation (19) by 
the system 
d. d 
(20) epee Se Ae, eee di 
Wet+b’y aut by 
where ¢ is an auxiliary variable introduced for the sake of symmetry. We have 
seen above (§ 59) that this system can be reduced to a simple canonical form 
by replacing x and y by two linear homogeneous combinations XY and Y of these 
variables. In this case the characteristic equation is 
s?— (av + b)s + ba — al’ = 0. 

This equation cannot have zero for a root, since we suppose that ab’ — ba’ is 
not zero. Several cases are now to be distinguished according to the nature of 
the roots: 

1) If the characteristic equation has two real and distinct roots 81, s3, we can 
reduce the system (20) to the form 

Ane Uk 


= ——= dt, 
$1 X SatY. 


and the given equation consequently becomes 
7 Vode “dy. 
S2 


The general integral is given by the equation 


59 


Vee Se 
Tf s; and sg have the same sign, Y approaches zero with _X, and all the integral 
curves pass through the origin, which is a node. If s2/s; is negative, there exist 
only two integral curves passing through the origin, the straight lines X = 0, 
Y=0; hence the origin is a saddleback. 
2) If the characteristic equation has two conjugate imaginary roots @ + fi, 
a — Bi(8 # 0), we can reduce the system (20) to the form 


d(X + Yi) PARAL Say XU a 

(a + pi)(X+ Yi) (a—Bpi)(X— Yi) © 

where X and Y are linear homogeneous combinations of « and y with real 
coefficients. We can then write these equations in the form 


EX HLs ae atid 
aXi— BY -pX + aY 


’ 


t, 


from which we derive 
XdAX+YdY XdY— YadAX 


a(X?4¥2), p(x? +4 ¥%)— 


180 NON-LINEAR DIFFERENTIAL EQUATIONS _ [Iv, § 65 


The general integral of the equation (19) is therefore represented by the 


equation 
a 
— arc tan 


Vx24+ Y2=0e8  * 


If @ is not zero, all these curves have the form of spirals which approach 
the origin as an asymptotic point. The origin is said to be a focus. 

If a is zero, the general integral is represented by concentric conics. The 
origin is called a center, but this case must be considered as exceptional, since it 
occurs only when a satisfies an equality. iN, 

8) If the characteristic equation has a double root s, that root is real and 
different from zero, and the system (20) reduces to the form 


dX dY 


— = —— = dd. 
sxX 38s(X+4Y) 


The equation (19) itself becomes dY/dX = 1+ Y/X, and the general integral is 
Y= CX +X Log X. In order to construct these curves, we can express X and 
Y in terms of an auxiliary variable by putting X = e8, which gives Y= Ce® + 6e9. 
When @ approaches — 0, X and Y, and consequently x and y, approach zero, 
and the origin is again a focus. 

The preceding classification is due to Poincaré, who has extended the dis- 
cussion to equations of the general form (14) whose coefficients are real. 


II. A STUDY OF SOME EQUATIONS OF THE FIRST ORDER 


66. Singular points of integrals. The developments in series by 
which we have established the existence of analytic integrals of a 
system of differential equations enable us to calculate these inte- 
grals only in the interior of the circle of convergence; but the 
knowledge of these developments suffices, as we have noticed in 
general (see II, Part I, § 86), to virtually determine these functions 
in the whole domain of their existence. Let us consider, for defi- 
niteness, an algebraic differential equation of the first order, 


(21) F(a, Y; y')= 0, 


where F is a polynomial in a, y, y'. Let (a,, y,) be a pair of values 
for which the equation F(a,, y,, y') = 0 has a simple root y. When 
«x and y approach 2, and y, respectively, the equation (21) has one 
and only one root approaching yj, and that root, y'= f(x, y), is a 
regular function in the neighborhood of the point (x,, y,). The 
equation (21) has therefore an analytic integral which reduces to 
y, for « = x,, and whose derivative takes on the value yj for x = z,. 
This integral is defined by its power-series development only in the 
interior of a circle C, about x»as center, whose radius is in general 
finite. But this function, whose analytic extension may be followed 


Iv,§66] SOME EQUATIONS OF THE FIRST ORDER 181 


outside of the circle C,, satisfies the equation (21) in its whole domain 
of existence. Let us observe that we may make use of the equation 
(21) itself to calculate the coefficients of the different series which 
we use in the method of analytic extension. If at a point a, in the 
circle C,, the integral considered is equal to y,, its derivative is equal 
to one of the roots y, of the equation F(a,, y,, y') = 0, and we shall 
be able to derive the values of the other derivatives at the point z, 
by successive differentiations of (21). 

It follows that every differential equation of the first order defines 
an infinite number of analytic functions, depending upon one arbi- 
trary constant. These are in general transcendental functions which 
cannot be expressed in terms of the classic transcendentals, and the 
same thing is true a fortiori of the functions defined by algebraic 
differential equations of the second, or higher, order. The study 
of the properties of these new transcendentals and their classifi- 
cation constitutes the object of the analytic theory of differential 
equations. 

We may be guided in this study by two different motives: we 
may seek the necessary and sufficient conditions that equations of a 
given sort may be integrated by means of functions already known; 
or, on the other hand, we may propose to ourselves the problem of 
discovering the algebraic differential equations that define transcen- 
dentals not reducible to the classic transcendentals, and possessing 
certain remarkable properties, such as being single-valued and ana- 
lytic, or analytic except for poles, etc. Whatever may be the object 
that we have specially in view, the investigation of the possible 
singularities of the integrals is an essential question. While the 
singular points of the integrals of a linear equation are fixed, the 
singular points of the integrals of non-linear equations vary in gen- 
eral with their initial values. For example, the integral of the equa- 
tion «+ yy'= 0 which takes on the value y, for «=0 is y= Vy — a”. 
This function has the two critical points + y,, — y,, which depend 
upon the initial value. Similarly, the integral of the equation y'= vy’ 
which takes on the value y, for «= 0 is y,/(1 — ay,); this solu- 
tion has the pole x =1/y,. We are therefore led to distinguish two 
classes of singular points for a differential equation: the fixed sin- 
gular points which do not depend upon the chosen initial values 
(not being necessarily singular points for all the integrals), and 
the movable singular points, poles, or critical points which depend 
upon the initial values. A differential equation may have both kinds 
of singular points. 


182 NON-LINEAR DIFFERENTIAL EQUATIONS _[iv, § 67 


67. Functions defined by a differential equation y' = R(x, y). We 
shall study in particular the differential equation 


(22) Y= R(x, aa 


where P(a, y) and Q(a, y) are two polynomials in a and y which 
have no common polynomial divisor. The pair of simultaneous 
equations P(«, y) = 0, Q(x, y) = 0 has a certain number of solutions 
(a,, 5,), +++; (Gn) 5). Let us mark the points a,, a,,---, a, im the 
x-plane. 3 

The transformation y = 1/z reduces the equation (22) to an equa- 
tion of the same form, 


dz Pita) 


(23) se aS Ra, 2) a Q(a, 2) ’ 


and the pair of equations P,(x, #)=0, Q,(#, #) =0 has a certain 
number of systems of solutions (a;, 0;),---, (@,, 9;,). Let us mark 
the points aj, a3,---, a, in the x-plane. The points a,, a, are in gen- 
eral singular points for some of the integrals of the equation (22), 
but they are known a priori; that is, they are the fixed singular 
points. : 

Let now (a, y,) be any pair of values such that Q(@,, y,) 1s not 
zero. Then by Cauchy’s fundamental theorem the equation (22) has 
an analytic integral, in the neighborhood of the point x, which 
takes on the value y, for « = x,. Suppose that we make the variable 
x describe any path Z proceeding from the point x, and not passing 
through any of the points a,;, az. We can continue the analytic 
extension of this integral along Z so long as we do not encounter 
any singular point. But it may happen that we are stopped by the 
presence of such a point; let @ be the first singular point which 
we encounter. The integral considered is analytic in the neighbor- 
hood of every point X of the path LZ included between &, and a, but 
the circle of convergence of the power series which represents it, 
and whose center is at X, never contains the point @ in its interior, 
however small |X — a| may be. The equation Q(a, y) = 0 has a cer- 
tain number of roots B,, B,,---, By. Let us mark the points £; in 
the y-plane. The equation Q(a, y) = 0 has only a finite number of 
roots, for otherwise the polynomial Q(a, y) would be divisible by 
(x — a) and the point @ would be included among the points a,, a;. 
For the same reason it is seen that the two equations P(a, y) = 0, 
Q(a, y) = 0 have no common root. | 


IvV,§67] SOME EQUATIONS OF THE FIRST ORDER 183 


There are now several possible cases to be examined. Let Y be 
the value of the integral in terms of X; we cannot suppose that Y 
approaches a finite value 6 different from B,, B,,---, By as X ap- 
proaches a, for R(x, y) is a regular function for « = a,y = B. Now, 
by Cauchy’s fundamental theorem, there would then exist a single 
integral approaching 8 as x approaches a, and that integral would 
be analytic at the point a, contrary to hypothesis. Let us suppose 
next that Y approaches the value B; when |X — @| approaches zero. 
The function R (a, y) is infinite for « = a, y = B;, but its reciprocal 
is a regular function, since we cannot have P(a, B;)=0. We have 
seen in § 63 that the equation (22) has one and only one integral 
which approaches 8; as |X — @| approaches zero, and which has the 
point @ as an algebraic critical point. Similarly, if | Y| becomes 
infinite as |X — a| approaches zero, the equation (23) has an integral 
which approaches zero with |X —a|. We cannot have simultane- 
ously P,(a@, 0) = 0, Q,(a, 0) = 0, since the point @ is not contained 
among the points az. If Q,(a, 0) is not zero, z is analytic in the 
neighborhood of the point a, which is a pole for the integral con- 
sidered. If Q,(a, 0) = 0, the point @ is an algebraic critical point 
for z and thus also for y. 

We have not yet exhausted all the possibilities. Might it not hap- 
pen, for example, that Y does not approach any limit, although | V| 
does not become infinite as |X — a| approaches zero? Painlevé has 
shown, in the following way, that this is not possible. Previously 
this had-been assumed without adequate proof. About the point a 
as a center let us describe a circle C with a very small radius «, The 
roots of the equation Q(X, y)=0 which approach respectively 
B,, By»-++, By as |X —a@| approaches zero remain respectively con- 
tained in the interior of the circles y,, y,,---, yy about the points 
By» By +++, By a8 centers with radii p,, p,,--+,; py. We can take the 
radius 7 so small that all these radii p; are themselves smaller than 
any given positive number e. Let us consider at the same time a 
circle T with a very large radius R, described in the plane of the 
variable y about the origin as center, and let (£) be the portion of 
the y-plane exterior to the circles y; and interior to the circle T. We 
shall show that when |X — a@| approaches zero, the corresponding 
point Y finally remains constantly in the interior of one of the cir- 
cles y; or exterior to the circle T. If this were not the case, we 
should always find on the path Z certain points X such that |X — a| 
is less than any given number and for which Y would be in the 
region (EZ). Suppose now that we have |X — a| < 7/2, for example, 


184 NON-LINEAR DIFFERENTIAL EQUATIONS _ [IVv, § 67 


while only values of Y in the region (£) are considered. We shall 
show that there is a positive minimum for the radius of the circle of 
convergence of that integral of the equation (22) which is equal to Y 
for x = X. In fact, there is evidently a maximum for |R (X, Y)| when 
the points X, Y remain respectively in the preceding regions, while 
there is a positive minimum for the numbers a and 6 (see § 22). Let 
7 be the minimum of the radius of this circle of convergence. We 
could find, by hypothesis, a point X' on the path ZL whose distance 
from the point a would be less than 7 and such that the correspond- 
ing point Y' would be in the region (£). Since the circle of conver- 
gence of the series, which represents the integral which takes on the 
value Y' for « = X', has a radius at least equal to y, the point a@ 
would be in the interior of this circle, which is evidently impossible, 
since @ is a singular point. 

The point Y, therefore, finally remains constantly in the interior 
of one of the circles y,; or outside of the circle Tas |X — «| approaches 
zero. Since the radius p; can be taken as small as we please and the 
radius R as large as we please, this means that Y approaches one of 
the values £, unless | Y| becomes infinite. We have just examined 
what happens in these two cases. It follows that the point @ is either 
a pole or an algebraic critical point. Hence we can avoid the singular 
point by replacing the portion of the path Z near the point @ by an 
are of a circle of infinitesimal radius described about this point 
as center, and we shall be able to continue the analytic extension 
beyond this point until we meet a new singular point. We shall 
show that on a path L of finite length there are never more than 
a finite number of poles or of algebraic critical points. In fact, with 
each of the points a;, a; as centers let us describe a very small circle 
in the plane of the w’s, and let us describe also a circle of very large 
radius about the origin as center, so that all of the path Z shall he 
in the region (E') of the x-plane bounded by these circumferences. 
Let x, be any point of (Z'). Then the integral whose absolute value 
becomes infinite as |~ — x,| approaches zero is equal to a polynomial 
in (« —«,)~* plus a power series in (# — a,) which converges in 
a circle of radius p,. Similarly, the different integrals which have 
the point x, as an algebraic critical point are represented by series 
arranged according to fractional powers of «—a,. Let p, be the 
smallest of the radii of convergence of these different series. It is 
clear that these numbers p, and p, vary continuously with the position 
of the point x,; hence they have a minimum d > 0, and the distance 
between two neighboring singular points on the path LZ is of necessity 


Iv,§67] SOME EQUATIONS OF THE FIRST ORDER 185 


greater than A.* We can therefore encounter on this path only a 
finite number of poles or of algebraic critical points. Consequently, 
the only movable singular points of the integrals of the equation (22) 
are poles or algebraic critical points. These integrals cannot have 
movable essentially singular points, and consequently they can have 
no natural boundaries. } 

The preceding arguments can be extended without difficulty to 
equations of the form (22), where P(a, y), Q(a, y) are polynomials 
in y whose coefficients are analytic functions of z. We have only to 
add to the points a,, aj, which are defined as above, the singular 
points of all these coefficients. If the path described by the variable x 
remains in a region not containing any of the points a;, a; or any 
of the singular points of the coefficients of the various powers of y, 
the only singular points which the integrals can have are poles or 
algebraic critical points. 

As an application, let us consider the question of finding the 
equations of the form (22) which have no movable critical points. 
In order that this may be true, the denominator must not contain y. 
In fact, let @ be any value of x, and B a corresponding value of y, 
for which Q(a, 8) = 0, while the numerator P (a, 8) is not zero. The 
integral of the equation (22) which approaches B when |x —a| 
approaches zero has this point as a critical point, and it is clear 
that it is not a critical point for all the integrals. Hence the desired 
equation must be of the form 


dy 
Fe Pay + Pray toes 
where P,,, Pm—1,°:: are functions of x Moreover, the equation 


obtained by putting y= 1/z must be of the same form, so that m 


* It should be noticed that an integral can have an infinite number of critical points, 
and even an infinite number of them in the neighborhood of any value of x. Consider, 
for example, the equation 

2yy’ = R(x), 


where R(x) is a rational function; the general integral of this equation is 


x 
y2=Ynt R(x) dx. 
Xo 
Let us suppose that the definite integral ih R(x) dx has the four periods 1, @, i, Biz, 
where @ and £ are two real irrational numbers. In the interior of a circle c described 
with any point x, as center and with an arbitrary radius, it is easy to prove (see II, 
Part I, §53, Note) that we can find an infinite number of roots of y? by suitably 
choosing the paths of integration, and each of these roots is a critical point. But 
a path of finite length described by the variable never contains an infinite number 
of them. 


186 NON-LINEAR DIFFERENTIAL EQUATIONS _[rtv, § 67 


cannot be greater than 2. It follows that the most general equation 
which satisfies the given conditions is a Riccati equation. It is 
easily verified that the condition is satisfied by any Riccati equa- 
tion. If we take, for example, the expression in (26) (§ 40) which 
represents the general integral, it is clear that that integral can have 
for singular points, besides the singular points of the functions y,, 
y,, Only poles resulting from the roots of the denominator y, + Cy,, 
that is, poles which vary with the constant C. 


Similarly, we may consider the question of determining the equations of the 
form (22) whose integrals have no movable poles. Let m and n be the degrees 
of P(x, y) and of Q(z, y) with respect to y; the equation obtained by putting 


y = 1/z is of the form 
(24) C2 _ nt a—m ii), 
dx Q, (2, 2) 

where P, and Q, are two new polynomials in z. Let =a be any value of z 
not contained among the fixed singular points. The equation (24) has an inte- 
gral which approaches zero as |* — a@| approaches zero. It would seem from 
this that the equation (22) always has an integral whose absolute value becomes 
infinite as |~ — a@| approaches zero, but this conclusion is incorrect if the inte- 
gral of the equation (24) reduces to z=0. It is necessary and sufficient for 
this that m < n+ 2; hence this is the condition that there shall be no movable 
poles. It follows that the only type of equation which has no movable singular 
point of any kind is the linear equation. 


Application. The preceding result enables us to determine whether the gen- 
eral integral of a differential equation of the first order is a rational function 
of the constant of integration, when that constant is suitably chosen. Let 


P(x, C) 
Q(z, C) 
be a rational function of the parameter C, where the coefficients of the two 


polynomials in C, P(z, C) and Q(z, C), are any functions of x. It is clear that 
the derivative 7’ is also a rational function of C, 
} 


(A) y=, C)= 


Y it (tC), 
. The elimination of the parameter C leads to a relation of the form 
(E) Fiy, y’; 2) =9, 


where F is a polynomial in y, y’ whose coefficients may be any functions of a. 
From the manner in which this equation is obtained we see that it is of defi- 
ciency zero in y and y’, regarding x as a parameter. 

Conversely, let us consider a differential equation of the first order (E), in 
which the left-hand side is a polynomial in y and 7’ whose coefficients are any 
analytic functions of x. In order that such an equation have a general integral 
of the form (A), it must first of all be of deficiency zero in y and y’, When 
this is the case, we can express y and 7’ as rational functions of a parameter u, 


Y= r(x, u), y= ry (x, u), 


Iv,§68] SOME EQUATIONS OF THE FIRST ORDER 187 


in such a way that we have, conversely, u = s(x, y, y’), where the functions r 
and r, are rational in u, and where s is a rational function of y and y’. Hence 
the given differential equation (E) may be replaced by the equation 


ér ordu _ 1, (a, U) 
one owas aanverTan 
which is of the form 
du 
(E,) FIsa F(x, u), 


where F is a rational function of u. If the general integral of the equation (E) 
is y= R(a, C), the general integral of the equation (E,) is, according to the 


above 
u = s[x, R(x, C), R’(z, C)); 


that is, it is a rational function of C. But the only singular points of such an 
expression which vary with C are evidently poles. The only movable singular 
points that the equation (E,) can have are therefore poles ; consequently the 
equation (E,) must be a Riccati equation.* 

Let us consider, for example, the equation 


y® = (Py + Q)? (y— a) (y— 9), 
where P and Q are functions of z, and where a and 6 are two constants. This 
relation is of deficiency zero in y and y’, and in order to express y and 7 as 
rational functions of a parameter, we need only set (y — b)/(y — a) = t?, which 
ce y (1-2)? = 6 — a) [Pot — at?) + Q¢-#)), 
and the equation (E,) is the Riccati equation 


2= = P(b— at”) + Q(1—#). 


68. Single-valued functions deduced from the equation (y')” = R(y). 
Let us now consider the integrals of the differential equation 


f N\n — ae PY) 

(25) (y') soit) — Q(y)’ 
where m is a positive integer and where P(y) and Q(y) are two 
mutually prime polynomials in y with constant coefficients. We shall 
now propose to determine all the equations of this kind whose gen- 
eral integrals are single-valued and in general analytic. Let x, be 
any value of a, and y, an arbitrary value of y which does not cause 
either of the polynomials P(y), Q(y) to vanish. The equation (25), 
after y has been replaced by y,, has m distinct roots in y'. Let us 
choose one of these. roots, yy. The equation (25) has an analytic 


* The converse is immediate. If (EK) is a Riccati equation, the general integral u 
is a linear function of the arbitrary constant C, and consequently y=r(x, u) is a 
rational function of C. 


188 NON-LINEAR DIFFERENTIAL EQUATIONS _ [iv, §68 


integral in the neighborhood of the point a,, which takes on the value 
y, at this point and whose derivative is equal to y, fora2=a,. We 
can extend this integral analytically along the whole of any path 
starting from the point x,, so long as we do not encounter any singu- 
lar points. Let @ be the first singular point encountered, let X be a 
point of the path Z lying between a, and a, and let Y be the corre- 
sponding value of the integral at the point X. The arguments of 
the preceding paragraph, which apply here without essential modi- 
fication, show that as |X — a| approaches zero, Y approaches a root 
of one of the equations 


PYy=9, Qy)=9, 


or else |Y| becomes infinite; but it can never happen that Y does 
not approach any limit. 

Let us examine the different possible cases. Suppose first that 6 
is a q-fold root of the denominator Q(y)= 0. From the equation (25) 
we have 


(26) de =(y—H)"[qte(y—b)+--Jdy. (4 #0) 


If y describes a path from y, to d in the plane of the y’s, the vari- 
able « describes a path starting from a, and ending at a point which 
we Shall call @ in the finite portion of the x-plane. Conversely, if x 
goes from x, to a along this path, y goes from y, to b. Putting 
y—b=t", we derive from the equation (26) a development of 
x—a in powers of ¢ beginning with a term of degree m+ 4q. 
Conversely, we have for ¢ a development according to fractional 
powers of x —a beginning with a term in (# — a)"™*®, and there- 
fore a development of y — 6 of the form 


ida pe 
y—b=(@—a)"*4[a,+a,(@—a)"tit...J.  (a,#0) 


Since g is positive, m + q is greater than m, and the point x =a is 
an algebraic critical point for the integral considered. In order that 
the general integral of the equation (25) may be a single-valued func- 
tion, it is necessary that the polynomial Q(y) shall reduce to a 
constant, or that the equation shall be of the form 


(27) (y')" = Py), 


where P (y) isa polynomial. Since the equation z™ = (—1)"2*"P (1/z), 
obtained by the transformation y =1/z, must also be of the same 
form, the degree of the polynomial P(y) cannot be greater than 2m. 


Iv,§68] SOME EQUATIONS OF THE FIRST ORDER 189 


Finally, we may suppose that P(y) 1s of degree 2m. In fact, if 
P(y) is of degree 2m — q, putting y=a-+41/z, where a is not a 
root of P(y), we are led to the equation 


em = (— 1)™[22"P(a) + 29-1 P l(a) + -- J, 
where the right-hand side is now a polynomial of degree 2 m. 

Conversely, given an equation of the form (27), where P(y) 
is a polynomial of degree 2m, if 6 is a root of that polynomial 
the substitution y=b+1/z leads to an equation in z of the 
same kind, where the right-hand side is a polynomial of degree 
less than 2m. ; 

Let us suppose, then, that P(y) is a polynomial of degree 2 m, and 
let a be the first singular point that we find on the path Z starting 
from x,. If |¥| becomes infinite when X approaches a, the equation 
in 2, obtained by putting y = 1/z, has an analytic integral which is 
zero for X = a. The point a is therefore a pole for y. 

There remains the case where Y approaches a root 6 of P(y) as X 
approaches a. This can happen only in case the order of multi- 
plicity of that root is less than m. For let us suppose that P(y) 
is divisible by (y — 6)’, where g = m. From the equation (27) we 
obtain, according to the initial conditions, 


i 
ae 8 ody, 
Vo (y — b)™ 


where #(y) is regular in the neighborhood of the point 6, and we 
see that |X| becomes infinite as Y approaches J. It follows that 
q < m, and the given equation can again be written in the form 


(8) \ de=(y—b) "[qte(y—b+--Jdy, (#0) 


whence we may derive a development of « — @ in powers of (y — b)”™ 
beginning with a term of degree m — g. Conversely, we may derive 
from (28) a development for y — 6 according to fractional powers of 
xz —a beginning with a term in («—a)”"—®, The point @ is there- 
fore, in general, an algebraic critical point. In order that it may be 

|_an ordinary point, m/(m — ¢) must be an integer 7; that is, we must 
have g =m (1 — 1/7), where 7 is an integer greater than unity. This 
condition is also sufficient, for if it is satisfied, we may derive from 
the equation (28) a development of the form 


: oe | 
%—a=k(y—b)'+k(y—b)* +::-, (k, # 9) 


190 NON-LINEAR DIFFERENTIAL EQUATIONS | [Iv, §68 


and, conversely, we have for (y — b)“ a eae according to 
integral powers of x — a. 

In order that the integrals of the equation (27) shall have no 
critical points, where P(y) is a polynomial of degree 2 m it 1s neces- 
sary and sufficient, by what precedes, that the order of multiplicity 
of each root of P(y)= 0 be equal to or greater than m, or of the form 
m(1—1/i), where i is an integer greater than unity. If all these 
conditions are satisfied, the general integral of the equation (27) is 
a single-valued function whose singular points in the finite portion 
of the plane can only be poles. 

To complete the discussion, we shall distinguish several cases : 

First case. There is one linear factor in P(y) whose exponent is 
greater than m (evidently there can be only one). If there are also 
p linear factors distinct from this one, the sum of the exponents of 
these factors is less than m: 


m(1—7)4 ine 
Y Uy 


Hence we have p—1<1/i,+.---+1/i, and, since i,,7,,---, 4, are 
each greater than unity, p —1 <p p/?, or p< 2. We have therefore 
p =1, and, extracting the mth root of the two sides, we may write 
the equation (27) in the form 


1 ay 

() y=Ay—a) yd) 
The case where i =1 should not be excluded, for it corresponds to 
an hypothesis which we have not examined — that of a single linear 
factor in P(y). 

Second case. The equation P(y) = 0 has an m-fold root. If it has 
two, the equation (27) becomes, after extracting the mth root of the 
two sides, 


(11) yi =A ty — a) (y — 8). 


If the equation P(v) = 0 has only one root of multiplicity m, it has 
p(p = 2) roots whose order of multiplicity is less than m, and we 
have a relation of the form 


or 


whence we derive p = 2, 


IV,§68] SOME EQUATIONS OF THE FIRST ORDER 191 


Since p is greater than unity, we have necessarily p = 2, i, = 7, = 2; 
the number m is an even number, and the equation (27) reduces 
to the form 


(IIT) Y= Ay ey ey 6), 
a, b, ec being three different numbers. 

Third case. The equation P(y) = 0 has only roots whose order of 
multiplicity is less than m. Let p be the number of these roots; the 


sum of the orders of multiplicity being equal to 2m, we have a 
relation of the. form 


(1 — 7)+ m(1 — -)+ eo m(A — >)= 2m, 
Be 4 % 


or 


Hence p = 4, and since p > 2, we can have only p= 4 orp=3. If 
p =A, the sum 1/1, +1/1,4+1/i,+1/1, must be equal to 2; and 
since each of the denominators is equal to at least 2, we must have 


pias TI Ge pee ae Bie ean 9 
Rifas Ua U0 me ie 


If p = 8, it is a question of finding three integers, i,, 7,, 4,, each 
greater than unity, such that the sum of their reciprocals is equal 
to 1. If none of these numbers is equal to 2, each must be equal to 
3. If one of them, 7,, is equal to 2, the sum of the reciprocals of the 
other two must be equal to 1/2; if the two are equal, each of them 
is equal to 4. If they are unequal, the smaller must be less than 4; 
it is therefore equal to 3, and the greater is then equal to 6. We 
have, then, in all only four possible combinations, and the equation 
(27) may be reduced to one of the following forms: 


(IV) y®=Aly—a(y—dy-—O)y—4), 
(V) Ya ayy 6) a — ey, 
(VI) yo = Ay ayy — 0) yey’, 
(VII) Yi aaa (hb) are), 


where a, 0, ¢, d are different numbers. If, in the equation (27), the 
polynomial P(y) is of degree 2m, and if the general integral is a 
single-valued function, the equation (27) has one of the forms which 
we have just obtained. Conversely, every integral of any one of 
these equations is a single-valued function, since on any path 
described by the variable we cannot encounter any other singular 
points than poles, 


192 NON-LINEAR DIFFERENTIAL EQUATIONS | [Iv, §68 


It is convenient to add to the types which we have just enumer- 
ated, in order to have all the equations of the form (25) whose 
general integral is single-valued, the types which are obtained by a 
transformation of the form y — a=1/z, where @ is a root of the 
polynomial P(y). The new types of equations thus obtained are 


(a) y'=A(y—a) ° 

(I)" y'=A(y—a) 4, 

(Gab) y'=A(y—@), 

(IIT)! y" = A(y — a)'(y — 5), 
(IIT)" y® = A(y— 6) y—2), 
Ey), y® = A(y— a) y—b)(y—°), 
(Vv)! ye — A (y a8 a)? (y eas b)’, 
(VI)! y* = Ay —a)*(y — 4)’, 
(VI)" ye — A (y pds a)? (y ot b)*, 
(VIT)’ y= A(y— a)(y — bf, 
(VALU) y" sn A (y Bad, a)? (y ae oye 
GVLI) Yt ’ ae = as (y Ase a)* (y nem | b)*. 


The equations (I), (1)', (I)", which are transformable one into the 
other, have a rational function for their general integral, as we see 
immediately from the equation (I)', for example. It is easy to show 
that the equations (II), (II)', (III), (1ID)', (1IT)" have a simply 
periodic function for their general integral. Finally, the general 
integral of the equations (IV) and (IV)' is an elliptic function. 
There remain, then, as new types of differential equations of the 
form (25) whose general integral is single-valued, only the equations 
(V), (VI), (VII), and those which reduce to these forms. These 
equations separate into three groups, and it is sufficient to integrate 
one equation from each of the groups, for example, the equations 
CV NV Ly 2 Weld ae 

If, in the equation (VI)"", we put y =a + 2? and extract the square 
root of the two sides, it becomes 


Ag? — Ate (2 +a—b), 


and the general integral in z is an elliptic function. Similarly, if in 
the equation (VIT)'" we put y=a-+ 2* and extract the cube root 
of the two sides, it becomes 


Sesh ee At (2 +a—b), 


which is an equation of the form (IV)'. 


Iv,§68] SOME EQUATIONS OF THE FIRST ORDER 193 


In order to integrate the equation (V)', we observe that that rela- 
tion between y and y' is of deficiency one. We can therefore express 
y and y' rationally in terms of the codrdinates of a point of a cubic 
or in terms of a parameter ¢ and the square root of a polynomial of 
the third degree. In fact, if we put y'= At’, we derive from the 


equation (V)! 
ya Gas oat, 


and the relation dy = y'dx leads to the new equation 
3 = Va br +4 Ae. 


The general integral of this equation, ¢= f(a + C), is an elliptic 
function. Hence the general integral of the equation (V)! is of the 


form 
ty a 
y= +5 (@ + C). 


It follows that the general integral of every equation of the form 
(25), if it is a single-valued function, is a rational function of x or 
of the exponential function e*”, or is an elliptic function. 

Except in the cases which have just been enumerated, the general 
integral of the equation (25) is never a single-valued function. For 
example, the inverse function of a hyperelliptic integral of the first 
kind cannot be a single-valued function. In fact, let P(vy) be a poly- 
nomial prime to its derivative and of degree greater than 4. The 
differential equation y” = P(y) cannot have a single-valued integral. 
Let (x,, y,) be the initial values of the two variables x and y. As 
|y| becomes infinite, « approaches a finite value @; and, conversely, 
when x goes from a, to a, |y| becomes infinite. The point x = @ is 
an algebraic critical point, as we have just seen, for the integral of 
the equation 2” = z*P(1/z) which approaches zero when a approaches 
a, since the degree of P(y) is greater than 4.* 


* In one of their papers Briot and Bouquet set for themselves the problem of de- 
termining all the equations F (y, y’)=0, where F’ is a polynomial, whose general inte- 
gral is a single-valued function (Journal de l’ Ecole Polytechnique, Vol. XXI). From 
the conditions found by them Hermite proved that the relation between y and 7’ is 
of deficiency zero or one (Cours lithographié de l’ Ecole Polytechnique, 1873); hence 
we can apply the method of § 11 in integrating them. If the relation is of deficieney 
zero, We can express y and 7/’ as rational functions of a parameter ¢. In order that 
the integral of the given equation be a single-valued function, the variable x, which 
is obtained by a quadrature, must be a linear function of ¢, x=(at+ b)/(ct+d), or 
else the logarithm of such a function, x= A Log [(at + b)/(ct + d)]. If the relation is 
of deficiency one, we can express y and 7/’ as elliptic functions of a parameter u, and 
dz/du=(1/y’) dy/du must reduce to a constant. The problem of Briot and Bouquet 


194 NON-LINEAR DIFFERENTIAL EQUATIONS _[iv,§69 


69. Existence of elliptic functions deduced from Euler’s equation. The reasoning 
of the preceding paragraph proves, in particular, that the general integral of 
the equation y’? = R(y), where R(y) is a polynomial of the third or of the fourth 
degree, prime to its derivative, is a single-valued function analytic except for 
poles in the whole plane. On the other hand, the inverse function, which is an 
elliptic integral of the first kind, has two periods whose ratio is imaginary (see 
II, Part I, § 56). This single-valued function is therefore doubly periodic, and 
we thus demonstrate the existence of elliptic functions by means of the integral 
calculus alone. 

The preceding proof of the single-valued property of the inverse function of 
the elliptic integral of the first kind is distinct from the one which has been 
given in § 78 of Vol. II, Part I, in which we make use of the properties of the 
function p(u). We shall also show in brief how we can take as our point of 
departure the theory of the integration of Euler’s equation, which will give an 
idea of the method pursued by the originators of the theory. 

Let us first consider the differential equation 


dz dy 


i 


pease Ch ba sie 22 
Vi-g V1—y 


whose general integral is x Nee y+ty Vi-x2=0 (§ 14). It is also clear that 
the general integral is given by the equation 


(29) 


arc sing + arcsiny = C’, 
and therefore that we have between the two a relation of the form 
are sin + arc siny = F(eV1i— y2 + yV1— 22). 


In order to determine the function F, let us suppose y = 0; there results the 
definite relation 


(80) are sin @ + arc sin y = arc sin (a V1— y2 + yV1—2@?). 


This relation is equivalent to the addition formula. For let us take two angles 
u and v determined by the conditions 


© = Sint, V1— a? = cosu, ‘"y ==/Silr 0, V1— 7? = cos, 


where the radicals have the same values as in the preceding equations. The 
relation (380) gives 

aV1—y+yV1— 2 = sin(u + v), 
or 

sin (wu + v) = sin wcos v + sin v Cos U. 


However, to see in this work only an ingenious proof of the addition formula 
for the sine function would be to overlook entirely its broad significance. 
Indeed, we shall show that it leads to a very simple proof of the existence of a 
single-valued integral function which satisfies the differential equation 


(31) y%@=1-y?, 


has been generalized by Fuchs, who formulated the necessary and sufficient condi- 
tions that the general integral of an equation of the first order F(x, y, y’)=0, alge- 
braic in y and y’, may have only fixed critical points. Poincaré has since shown that 
when these conditions are satisfied, we are led to quadratures or to Riccati equations 
(Acta mathematica, Vol. VII). ; 


Iv,§69] SOME EQUATIONS OF THE FIRST ORDER 195 


and which reduces to zero for x = 0, while y’ is equal to + 1forz = 0. Cauchy’s 
general theorem shows, indeed, that there exists an analytic function ¢ (2) satis- 
fying these conditions and analytic in the neighborhood of the origin, but it does 
not give the radius of convergence of the power series which represents ¢ (2). 
Let R be this radius of convergence. The circle C of radius R about the origin 
as center is the greatest circle described about the origin as center within which 
the function ¢ (x) is analytic. The derivative ¢’(z) is analytic in the same circle, 
and we have ¢7(x) =1— ¢?(x). Let us now return to the equation (29), and let 
us make the change of variables z = ¢(u), y = $(v), where wu and v are the two 
new variables and ¢ the function which has just been defined. If we choose 
the determination of the radicals in a suitable way, we have also 


V1—-x=¢(u), V1i—y=¢"(0), 


and the equation (29) becomes du + dv =0. The general integral of this equa- 
tion can therefore be written in two different ways: 


UP os CU, aVv1—ytyV1l—2=C’ 
or 
$(u) $’(v) + 4/(w) o(v) = C’. 
Hence it follows, as before, that we have a relation between u+v and 
p(u) p’(v) + p’(u) P(v). Putting v = 0, the relation is determined, and we have 


(32) p(u + 0) = H(U) o’(v) + ou) Ov). 
This relation holds, provided |u|] < R, |v]|< Rk, |u+ v|<R, which will surely 


be true if we have |u|< R/2, |v]|<R/2. Let us put v=u and |u|<R/2; then 
the equation (82) becomes 

(33) p(2u) = 2g (u) o’(u). 

Let ¢,(u) be the function 2 ¢ (u/2) ¢’(u/2). This function ¢,(u) is analytic in 
the circle of radius 2 R about the origin as center, and, by (83), it is identical 
with the analytic function ¢(u) in the circle C of radius R. These two func- 
tions, ¢(u), ¢, (uv), form therefore only a single analytic function, which is ana- 
lytic outside of the circle C. It is therefore impossible that the radius R of this 
circle of convergence has a finite value ; consequently the function ¢(u) is an 
integral function of u. 

Let us now consider the differential equation 

(34) y? = (1—y?) (1 — by’), 
adopting for the right-hand side Legendre’s normal form, and let us study the 
integral A(x) of this equation which is zero for « = 0 and whose derivative is 
equal to +1 forz=0. This function A(z) is analytic in the neighborhood of 
the origin. Let C be the greatest circle about the origin as center in the inte- 
rior of which the function (zx) is analytic except for poles, and let R be its 
radius. If the nearest singular point of (zx) to the origin were not a pole, we 
should take for C the circle through this singular point, and the function d(z) 
would then be analytic in this circle. 

Let us now consider Euler’s equation 


dz, dx, 


ee 
(L=a})(1— Haj) (1 — 3) (1 — Wa) 


=.(), 


(35) 


196 NON-LINEAR DIFFERENTIAL EQUATIONS _ [Iv, §69 


Multiplying the numerator and the denominator of the right-hand side of the 
equation (66) (p. 29) by the conjugate of the denominator, and suppressing 
the common factor z}— «3, we have for its general integral 


ty / (I= a) (1 Hei) + V2) 0-2) 


36 
G°) 1— aie 


On the other hand, choosing the sign of the radicals suitably, the change 
of variables x, = A(u), x, =d(v) reduces the equation (35) to the equation 
du + dv = 0, whose general integral is u+ v= C’. If we make the same sub- 
stitution in the formula (36), we have a relation of the form 


(wu) X (0) + A(W)®’(u) 
TOPOS Ree 


We can determine the form of the function F, as before, by supposing v = 0, 
which gives F(u) = A(u); and we have the definite relation 


X(u) NX (0) # ACL) XW) 


See ME) Ss SEAT) 
Putting v = u, we find 
_ 2r(u) ’"(u) 
(38) OY) — ea aayt 


a formula which holds true whenever |u|< R/2. 


2 2 


1— (3) 
2 


This function is analytic except for poles in the circle of radius 2 R about the 
origin as center, since it is the quotient of two such functions. Moreover, it 
coincides with \(u) in the interior of the circle C, by the relation (38). Hence 
the two functions A(u) and @(u) form a single analytic function, and )(u) is 
analytic except for poles in a larger circle than C. It is therefore impossible to 
suppose that the radius R of this circle has a finite value, and consequently the 
function \(u) is analytic except for poles in the entire plane. 

The equation (37) constitutes the formula for the addition of the arguments 
of the function \(u). When k approaches zero, we find again at the limit the 
addition formula for sinu. The function sin u can, in fact, be considered as a 
degenerate case of \(u), obtained by letting k approach zero. 


@(u) = 


70, Equations of higher order. The study of the properties of the functions 
defined by differential equations of higher order presents much more serious 
difficulties than those which arise in studying equations of the first order. These 
difficulties result in a great measure from the possible presence of movable 
essential singularities. These singularities may be at the same time essentially 
singular points and transcendental critical points, as in the following example, 
due to Painlevé. The function 


(89) y = p[Log (Az + B); go, 9], 


IV,§70] SOME EQUATIONS OF THE FIRST ORDER 197 


where A and B are two arbitrary constants, is the general integral of the equa- 
tion of the second order, 


g 
642 — 92 
PT ei 1 


(40) oat - be 
4° %Y— 9, V4y®— gay — gs 


In the neighborhood of every value of « different from — B/A this function 
(89) is analytic or analytic except for poles. 

When z turns around the point — B/A, the function has an infinite number 
of different values, unless 27i is an aliquot part of one of the periods of the 
function p(W; J, gz). On the other hand, when the variable z describes any 
sort of curve such that | Az + B| approaches zero, the point which represents 
the quantity u = Log (Az + B) describes a curve with an infinite branch. This 
curve therefore crosses an infinite number of parallelograms of periods of 
the function p(u), and consequently y does not approach any finite or infinite 
limit. Thus, although the general integral of the equation (40) presents neither 
fixed critical points nor movable algebraic critical points, we cannot conclude 
from this that it is single-valued. This is on account of the presence of the 
movable transcendental critical point, c= — B/A. 

Beginning with equations of the third order, the movable transcendental 
singular points may form lines. It is easy to see how an analytic function hay- 
ing natural boundaries may not have any critical points in its whole domain of 
existence without being on that account single-valued. Consider, for example, 
a function f(z) analytic in the ring included between the two circles C and C’ 
described about the center a and having C and OC’ as natural boundaries 
(II, Part I, § 87). The function F(x) = f(z) + Log (x — a) has the same circles 
C and:C’ as natural boundaries, and it is analytic at every point between C 
and O’, Nevertheless it has an infinite number of determinations for every 
value of z in this domain. 

For a long time these difficulties arrested the progress of this theory, but 
Painléve, in recent investigations, has obtained algebraic differential equations 
of the second order which are integrable by means of essentially new single- 
valued transcendentals. We shall cite only one of the equations given by 
Painlevé, a i 

y" = ay” + BE, 
where a and £ are constants (a8 40). The general integral of this equation is 
a transcendental function analytic except for poles.* (Bulletin de la Société 
Mathématique, Vol. XX VIII.) 


* Starting with linear equations, it is easy to form systems of differential equa- 
tions which generalize Riccati’s equation, and whose integrals have no other movable 
singular points than poles. Consider, for example, a system of three linear equations 
of the first order, 


(aq) yt+ayt+bz+cu=0, “7+ ayyt+b4z+c,u=0, U’ + day t+ bez+cgu=0. 
If we put y=uY, z=uZ, Yand Z are integrals of the system of equations 


YtaY+bZ+c— VY (a,V+ bgZ+ cg) =0, 


(6) Z’+a,V¥+b,Z+0e,-Z(a,¥t bg Z + Cg) = 0, 


and it is clear that the only movable singular points of the integrals are poles. But 


198 NON-LINEAR DIFFERENTIAL EQUATIONS _[iv,§71 
Tl. SINGULAR INTEGRALS 


71. Singular integrals of an equation of the first order. We have 
already remarked on several occasions ($§ 10, 14) that a differential 
equation of the first order may have certain integrals which it would 
be impossible to obtain by assigning a particular value to the arbi- 
trary constant which appears in the general integral. This result 
appears to contradict the theorem of § 26, from which we deduced a 
precise definition of the general integral. This leads us to consider 
again Cauchy’s fundamental theorem and to determine by a closer 
examination whether or not the hypotheses of that theorem are 
necessarily fulfilled for all integrals. Let us consider, for definite- 
ness, an equation of the first order, 


(41) , F@,y,y')=9, 
where F is an irreducible polynomial in a, y, y' of the mth degree 
in y'. To every system of values (a,, y,) the equation 


(41') FY» y') =9 

determines in general m corresponding distinct and finite values 
Yip Yar ***) Ym for y'. Let us suppose first that this is actually true 
at a given point («,, y,). Then, as «—a, and y — y, approach zero, 
the m roots of the equation (41) approach respectively yj, 43, +++) Yny 
and each of them is an analytic function in the neighborhood of the 
point («,, y,). The root which approaches y;, for example, is repre- 
sented by a power-series development of the form 


(42) y =yita(e—x%)+Bly—y) +e 


We can apply Cauchy’s theorem to the equation (42), and we con- 
clude from it that that equation has one and only one integral which 
approaches y, as |2 — a,| approaches zero. This integral is analytic, 
and the development of y — y, begins with the term y;(« —,). To 
each root of the equation (41') corresponds thus an integral of the 
given equation. 


it is to be noticed that this is not the most general system of differential equations 
of the form 


(7) Y=R(& Y,Z), 2 =R (x, Y, Z), 


where Fk and R, are rational functions of Y and Z, which possess this property. 
In fact, let Y=@(¥y, Z;), Z=Y (Yj, Z1) be relations defining a Cremona trans- 
formation, such that we can derive from them the inverse relations Y;= 9, (Y, Z), 
Z,= 1 (Y, Z), where ¢, Y, $1, Y; are rational functions. If we apply this trans- 
formation to the system (8), we are led to a system having the same property, which 
is surely of the form (y) but not in general of the form (8). 


iv, § 71] SINGULAR INTEGRALS 199 


The equation (41) has therefore m and only m integrals which 
take on the value y, for «=«,, and these m integrals are analytic 
in the neighborhood of the point z,. In geometric language we may 
say that through the point 1, whose codrdinates are (a,, y,) there 
pass m integral curves with m distinct tangents, and that the point 
M, is an ordinary point on each of them. Besides, all the integrals 
of the equation (42) which for =a, take on values differing only 
slightly from y, satisfy a relation of the form ®(a, y; x, y, + C) =0 
($ 26), and the integral considered corresponds to the value C = 0 
of the arbitrary constant. 

If for x=2,, y=y, a root of the equation (41’) is infinite, it 
will suffice to regard y as the independent variable and a as the 
dependent variable. The equation (41) is replaced by an equation 
of the same form, F(a, y, x,) = 90, which forx=a2,, y=y, has a 
zero root x! = 0. If this is a simple root, we derive from it a develop- 
ment for « — x, in powers of y — y, beginning with a term of at least 
the second degree. Conversely, the point x, is an algebraic critical 
point for the integral which approaches y, when |a — x,| approaches 
zero (II, Part I, §100). Through the point (#,, y,) there passes an 
integral curve whose tangent at that point is the straight line « = x, 

The coérdinates (a,, y,) of a point for which the equation (41) 
has a multiple root satisfy the relation 


(43) R(x, y) = 9, 


which is obtained by eliminating y’ from the two relations F = 0, 
0F/dy'=0. The equation (43) represents a certain curve (y), and 
for all the points of this curve the equation (41) has one or several 
multiple roots. Let (@,, y,) be the coérdinates of an ordinary point 
M, taken on this algebraic curve. We shall suppose, in order to 
treat the simplest possible case, that the equation 


Ce Ci eel 
has a double root yj; but no other multiple root finite in value. If 
this double root were infinite, it would suffice to interchange x and 
y in order to pass to the case where it is zero. When |a — a,| and 
|y—y,|are very small, the equation (41) has two roots which differ very 
little from yj. These roots are not, in general, analytic functions of 
the variables x and y in the neighborhood of the point (a,, y,), but 
their sum and their product are analytic functions,* so that these two 


* The proofs of these properties are analogous to the proofs of the corresponding 
theorems on implicit functions of a single variable (II, Part I, § 98). 


200 NON-LINEAR DIFFERENTIAL EQUATIONS _ [Iiv,§71 


roots of the equation (41), which approach yj as |a — x,| and |y — yo| 
approach zero, are also roots of an equation of the second degree, 


(44) y" —2P(a, yy + Q (a, y) = 9, 


where P(a, y) and Q(a, y) are analytic functions in the neighbor- 
hood of («,, y,). From the equation (44) we find 


(45) y' = P(«, y) EVP? (@, y) — A@, 9), 
and these two roots are equal for all the points of the curve (y,) 
whose equation is P? — Q= 0. This curve (y,) is necessarily part of 
the curve (y), and since it passes through the point («,, y,), 1t coin- 
cides with (y) in the neighborhood of (a, y,). In order to study 
the corresponding integral curve, we shall suppose that the origin 
has been transformed to the point J/,, which amounts to putting 
a, =y,=9. Since the origin is a simple point of the curve (y), if we 
have chosen the axes of codrdinates in such a way that the tangent 
at the origin is not the axis Oy itself, the equation P? — Q= 0 has an 
analytic root y = y,(@) which approaches zero as x approaches zero. 

In general, the slope of the tangent to the curve (y) at the origin 
is different from the double root 7, = P(0, 0) of the equation (45) 
fora=y=0. Let us first assume this point, which is almost self- 
evident, and return to it later. Then, if we put y=y,+ 2 in the 
equation (45), it becomes 


Yte=P(a,y, +2) + V2 (22), 
where ®(a, 2) 18 a power series in x and z. It is clear that 2 must 
be a factor under the radical after the substitution y = y, + 2, since 
y, is a root of the equation P?— Q=0. If we arrange (a, 2) in 
powers of z, we have a development of the form 


VAC) og zw, (x) oF zp, (a) Rue 

where y,, ¥,, ¥, are regular functions of x in the neighborhood of 
the origin. The function y,(#) cannot be zero for a = 0, for other- 
wise the development of z@(a, z) would contain no terms of the first 
degree in x, 2; whence the development of P? — Q would contain no 
terms of the first degree in a, y, contrary to hypothesis. Similarly, 
if we replace y, by its development in the difference P(a, y, + 2)— yj, 
we have, after arranging in powers of z, 


P(x, Y, aF z) oar A ve p, (2) ate zp, (x) se at a) 
where the first function ¢,(a) does not vanish for « = 0, since by 
supposition the derivative y;, is different from P(0, 0) at the origin. 


¥, $71] SINGULAR INTEGRALS 201 


The equation (45) therefore reduces to an equation of the form 
(46) z! = $,(“) a zp, (x) a Ve Vy (2) 19 aw, (x) cogs 
where neither of the functions ¢,(#) and y,(#) vanishes for x = 0. 


In this last equation let us put z = w*. Selecting a determination 
of the radical on the right, we find 


(47) au = >, (x) + uh, (e) + ++ fwVy, (a) + wy, (we) + -e-. 


The right-hand side is analytic in the neighborhood of the point 
a= 0, w = 0, since y,(0) is not zero. Moreover, this right-hand side 
is not zero for x = 0, wu = 0, since ¢,(0) is not zero. The derivative 
du/dx is infinite for « =w=0. Hence the equation (47) has one 
and only one integral which approaches zero as x approaches zero 
(§ 63), and for which the origin is an algebraic critical point. 

It follows that the given equation (44) has an integral y= y, + v? 
which approaches zero as x approaches zero. The adoption of the 
opposite determination of the radical in the equation (47) would 
amount to changing « to — w in that equation, and we should obtain 
the same function y, + wu. The origin is an algebraic critical point 
for this integral. Let a, be the term independent of 2 and of w in 
the development of the right-hand side of the equation (47), and let 
b, be the coefficient of w in the same development. Developing x in 


powers of wu, we find 
uw 2d, 


Fait a a 
a, 34% 


i 
Conversely, we derive from this a series for u in powers of x”, 
=e b 


and the development of y, + wv’ contains a term in 2°”, The origin 
is therefore a cusp for the integral curve which passes through this 
point, and we can say now that the curve (y), represented by the equa~ 
tion (43), is, in general, the locus of the cusps of the integral curves. 
Through a point of the curve (y) there passes, in general, an inte- 
gral curve that has a cusp of the first kind at this point, and the 
tangent at the cusp has for its slope the double root y.. If the equa- 
tion (41) is of higher degree than 2, there pass through the same 
point other integral curves, corresponding to the simple roots of the 
equation F (2); Yo) y') = 0, for which this point is an ordinary point. 
The discussion is entirely different when for every point (x,, y,) 
of the curve (y) the corresponding double root y; of the equation (41) 


202 NON-LINEAR DIFFERENTIAL EQUATIONS | [iv,§ 71 


is equal to the slope of the tangent to the curve (y) at this point. 
In this case we see first of all that the curve (y) is an integral curve 
of the equation (41). Moreover, it is an integral which is entirely 
unaccounted for in Cauchy’s fundamental theorem, whatever may 
be the point chosen on the curve to fix the initial values of x and y. 
For if we take the point (x,, y,), the equation 


F(x, y, y') = 0 

has two roots which approach yj as |« — ,| and |y — y,| approach 
zero; but these two roots are not in general regular functions of the 
variables 2 and y in the neighborhood of the values a, y,, and we 
cannot apply Cauchy’s theorem. The integral thus obtained is said 
to be a singular integral. The investigation of singular integrals 
does not offer any theoretical difficulties, since it is evidently suffi- 
cient to determine whether the curve represented by the equation 
(43) satisfies the differential equation (41), and this necessitates only 
an elimination. It may happen that the equation (43) represents two 
distinct curves, one of which is a singular integral curve and the 
other the locus of the cusps of the integral curves. 

If the curve (y) is a singular integral, through each point of that 
curve there passes in general another integral curve tangent to (y). 
Let us take for origin any point of (y). We know in advance an 
integral y, of the equation (45), namely, the singular integral for 
which we have simultaneously 


(48) yi = P(x, y,), P* (x, ¥,) = Q(a, Y,): 
Putting y = y, + 2, as above, the equation takes the form (46), but 
in this case the function ¢,(x) is zero, since z = 0 must be an inte- 
gral of this new equation. Retaining the other hypotheses, the func- 
tion y,(x) is not zero for « = 0, and if we next put ze =v’ in the 
equation (46), we are led to an equation all of whose terms are 
divisible by vw. Dividing by w, there remains a differential equation 


(49) 2u' = ul d(x) +w'g,(@) ++ TEVA) + wy, @) +++ 


to which we can apply Cauchy’s general theorem. Since the func- 
tion y,(a) is not zero for x = 0, the two determinations of the radi- 
cal are analytic for «= 0,uw=0. The equation (49) has therefore 
two analytic integrals in the neighborhood of the origin which van- 
ish for «= 0, and it is easily seen that these two integrals are 
deducible one from the other by changing uw to —wu _ It follows that 
the equation in y has another integral curve 


y¥=y,tu, 


IV, § 71] SINGULAR INTEGRALS 2038 


which is tangent to the curve (y) at the origin. But there is an 
essential difference between these two integrals. In fact, we can 
apply the general theorems of § 26 to the equation (49), and the 
integral of this equation which is zero for x = 0 belongs to a family 
of integrals which depend upon one arbitrary constant. The same 
thing is therefore true of the integral curve which is tangent to the 
singular integral curve at the origin, whereas the singular integral 
itself is in general an tsolated solution. This fact is easily explained, 
since we cannot apply to this integral the reasoning which proves 
the existence of a general integral (§ 21) from which we could 
obtain the former by giving a particular value to the constant 
which appears in the latter. 

The singular integral is therefore in general the envelope of the 
other integral curves. Lagrange had already noticed that the enve- 
lope of the curves represented by the general integral of a differen- 
tial equation of the first order is also an integral of the same equation, 
which is almost self-evident, since at any point of the enveloping 
curve the slope of the tangent is the same for the envelope and for 
the particular curve enveloped at that point. We can also find in 
this way the rule which enables us to deduce the singular integral 
from the differential equation itself. In fact, let us first take a point 
M very near the envelope. Through this point M there pass two 
integral curves very close to each other. Moreover, the slopes of 
the tangents to these two curves differ from each other very little. 
When the point M approaches the envelope, these tangents approach 
coincidence, and the equation (41) has a double root in y' (see I, 
§ 208, 2d ed.; § 202, Ist ed.). 

Summing up, we see that for an equation of the first order two 
entirely distinct cases may present themselves, according as the 
curve (y) is a singular integral curve or the locus of the cusps of the 
integral curves. It is natural to ask which of these two cases ought 
to be considered as the normal case. A little attention will show that 
it is the second. In fact, the curve (y) is also the envelope of the 
curves represented by the equation F(a, y, a) = 0, where a is the 
variable parameter. If the differential equation (41) had a singular 
integral, whatever the polynomial F might be, we should be led to 
assert a consequence which is manifestly absurd — that is, that at 
every point of the envelope of a family of algebraic curves the slope 
of the tangent is equal to the value of the parameter for the corre- 
sponding curve of the family tangent to the envelope at that point. 
If this condition is satisfied by a family of curves, it suffices to 


204 NON-LINEAR DIFFERENTIAL EQUATIONS | [Iv,§71 


change the parameter (putting, for example, a = a'+ e) in order 
that this condition shall cease to hold. We see, therefore, that if 
we start from an equation of the first order in which the coeffi- 
cients of F are taken at random, rather than from an equation fur- 
nished by the elimination of an arbitrary constant, the cases where 
there exists a singular integral must be considered as exceptional. 
If this result formerly appeared paradoxical to some mathemati- 
cians, that was no doubt because, up to the time of Cauchy’s work, 
the equations studied had been principally those whose general inte- 
gral is represented by algebraic curves. As a family of algebraic 
curves has in general an envelope, it appeared quite natural to 
extend the conclusion to the integral curves of any differential 
equation of the first order. We have just seen that this induction 
was not justified.* Moreover, even in the case where a family of 
plane curves depending upon a variable parameter has an envelope, 
the method which enables us to find that envelope gives also, as 
we have seen (I, §§ 207, 208, 2d ed.; §§ 201, 202, 1st ed.), the locus 
of singular points. 


72. General comments. Hxample 1. Let us take the equation 
(50) y? + 2ay'—y=0. 


The two values of y’ are equal for all the points of the parabola 
y +x? = 0, and the double root is equal to — 2, while the slope of 
the tangent to the parabola is — 2a. This curve is therefore not a 
singular integral curve. We shall show that it is the locus of the 
cusps of the integral curves. The equation (50) is a Lagrange 
equation. Applying to it the general method of § 9, we find that the 
codrdinates « and y of a point of an integral curve are expressed in 
terms of a parameter p by means of the equations 


(51) De ae gly ae 


* In the theory of envelopes we suppose tacitly that in the neighborhood of a 
system of solutions (x9, Yo, ao) of the two equations f(x, y, a) = 0, @f/da=0 the 
functions f and @f/da, together with their partial derivatives, are continuous, so that 
we can apply to the functions x and y of a defined by these two equations the reason- 
ing which we apply to implicit functions. Now, given a differential equation of the 
first order, we know certainly that it has an infinite number of integrals depend- 
ing upon an arbitrary constant and represented in a certain region by an equation 
¢ (x, y, C) = 0, but there is nothing to prove a priori that this function ¢ (2, y, C) 
satisfies the conditions which we have just mentioned. We may even assert that it is 
not true in general. oF! 


IV, § 72] SINGULAR INTEGRALS 205 


It follows that these integrals are represented by unicursal curves of 
the fourth degree. For the values of the parameter which are roots 
of the equation p*® + 3 C = 0 we have dx/dp = dy/dp = 0. Each of 
these curves has therefore three cusps, and the locus of these points 
can be found by eliminating p and C from the equations (51) and 
the relation p*® = — 3 C, which gives the parabola y + 2? = 0. 

Example 2. Let us again consider Euler’s equation Xy” = Y. The 
two values of y' are equal for all the points of any one of the eight 
straight lines represented by the equation XY=0. These eight 
lines represent the singular solutions, and form the envelope of the 
curves represented by the general integral. 

Kxample 3. We can use the following method to determine whether 
singular solutions exist. From what we have seen, such an integral, 
if it exists, satisfies the equations 


S 


F 

F(a, y, y')= 9, aiken? 
and consequently also the equation 0F'/éx + 0F /dy y' = 0 obtained by 
differentiating the first. Conversely, suppose that for all the points 
of a curve (y) the three equations 
ar Or , OF 


UPA Pah eh 


(52) F(x, y, m) = 0, a 


have a common solution in m. Along the curve (y), x, y, and m are 

three functions of a single variable satisfying the three relations 

(52). We have therefore the relation between their differentials, 
OF OF OF 


eee ats SN PC 


which, by (52), takes the form 


If éF/éy is not zero at all the points of the curve (y), we have 

therefore y' =m, and this curve is a singular integral curve.* If 

OF /éy = 0, we must also have 0F'/éx = 0, and a direct verification is 

necessary to determine whether the curve (y) is an integral curve. 
This remark apples in particular to Clairaut’s equation 


F(a, y, y)=fy', y — zy')= 0. 


* See an article by Darboux in the Bulletin des Sciences mathématiques, 
Vol. IV, 1873, pp. 158-176. 


206 NON-LINEAR DIFFERENTIAL EQUATIONS _ [iv,§72 


Putting, for the sake of brevity, w= y — ay’, the three equations 
which are to be compatible are here 


0 of af 
FU Wie Y= 01a oa eae te ac 


and they reduce to only two equations. A singular integral is there- 
fore obtained by eliminating y' from these two relations. 


Example 4. Consider the equation 
1 
a2 + y2— Qa(e+ WY) eg it ay) a, 


whose general integral is represented by the circles which have double contact 
with the conic x2(1— m2) + y2+K =0, 
and which have their centers on the z-axis. This conic represents a singular 
solution. Moreover, the two values of y’ become infinite for every point of the 
axis of x. This straight line is not, however, a locus of the cusps. Through any 
point of it there pass two integral curves tangent to each other, the common 
tangent being parallel to the axis of y. 

Example 5. In order that a curve C represent a singular integral, it is not 
enough to require that at all the points of that curve the equation (41) shall 
have a double root. It is also necessary that that double root shall be precisely 
the slope of the tangent to C. Let us consider, for example, the cissoids repre- 
sented by the equation (y— 2a)?(c—a)—z?=0. The straight line z= 0 is the 
locus of the cusps of these curves, and it represents also a particular integral 
obtained by supposing a= 0. At every point of this integral curve the corre- 
sponding differential equation has the double root y’ = 0 and an infinite root. 
It is therefore not a singular integral curve. 

Example 6. Let S be a surface having convex regions and also regions 
where its curvature is negative. These regions are separated by a curve I, the 
locus of the parabolic points, at every point of which the differential equation 
of the asymptotic lines (I, § 2438, 2d ed. ; § 242, Ist ed.), 


Ddu? + 2 D’du dv + D’dv? = 0, 


has a double root in dv/du. This double root furnishes the direction of the 
single asymptotic tangent. If the tangent to I does not coincide with this 
asymptotic tangent (which is the general case), the curve I is the locus of the 
cusps of the asymptotic lines; but if the asymptotic tangent at each point M 
of I coincides with the tangent to TI, the curve is the envelope of the asymptotic 
lines. This curve I, therefore, is at the same time an asymptotic line and a line 
of curvature, since the tangent is also an axis of the indicatrix. The normals 
to the surface S along I form, therefore, a developable surface, and since the 
normal to S is the binormal to the curve I, it follows that T is a plane curve 
(I, § 235, 2d ed.; § 231, 1st ed.) and the given surface S is tangent to the plane P 
of the curve T' along the entire length of that curve. 

Let us consider, for example, a surface of revolution. In order that one of 
the principal radii of curvature at a point M of this surface be infinite, the 
radius of curvature of the meridian must be infinite or the tangent to this 
meridian must be perpendicular to the axis. In the first case the curve T isa 


IV, § 73] SINGULAR INTEGRALS 207 


parallel each point of which is a point of inflection for the meridian, the asymp- 
totic tangent is perpendicular to the tangent to [, and this parallel is a locus 
of the cusps of the asymptotic lines, On the other hand, in the second case the 
curve I is a parallel in all of whose points the surface is tangent to the plane 
of this parallel, as in an anchor ring. It is also the envelope of the asymptotic 
lines. All these results are easy to verify directly from the differential equation 
of the asymptotic lines in polar codrdinates. 


73, Geometric interpretation. The preceding discussion may be presented in a 
somewhat different form, which we shall rapidly indicate. We shall continue 
to employ geometric language, although the reasoning can be extended without 
difficulty to the domain of complex variables, 

We have already pointed out (§ 8) that the integration of a differential 
equation of the first order F(x, y, y’) = 0 is equivalent to the determination of 
the curves I which lie on the surface S whose equation is 


(58) F(a, y, z)=0 


and for which dy — zdz = 0. The projection c on the zy-plane of a curve I of the 
surface S satisfying the preceding conditions is an integral curve of the given 
differential equation, and conversely. We shall suppose in the discussion that 
this surface S has no other singularities than the double curves along which two 
sheets of the surface cross with distinct tangent planes. Instead of studying 
the curves c in the zy-plane, we shall study the curves I on the surface S. 

Let us consider first a point Mj (%, Yo, 2) of the surface S not on a double 
curve nor where the tangent plane is parallel to the z-axis. The tangent to the 
curve I’ which passes through M, lies in the tangent plane at this point, 


(4 (X- a) (SF a) t =) (= Sy) t 22 (G) = 


and also, since we must have dy — zdz = 0, in the plane 
(55) Y — ¥) — %)(X — 2) = 0. 


These two planes are distinct, since (¢F'/éz), is not zero; hence they intersect in 
a straight line not parallel to Oz. Through the point M) there passes, therefore, 
one and only one curve I’ whose tangent is not parallel to the z-axis. The 
projection ¢ of this curve on the zy-plane passes through the point m,, the 
projection of M,, and m, is an ordinary point for c. If the point M, belongs 
to a double curve of S, the preceding reasoning applies to each of the two sheets, 
provided that none of the tangent planes at M, are parallel to Oz. Through the 
point M, there pass, therefore, two curves TI’ corresponding to the two sheets 
of the surface S. It remains to find out what happens if the point M, lies on 
the curve D of S, the locus of the points for which we have simultaneously 
F=0, éF/éz = 0. We shall suppose that this curve D is not a double curve. 
It is, then, the locus of the points of S where the tangent plane is parallel to Oz, 
and one at least of the partial derivatives 0F/dz, 6F/dy is different from zero 
at the point M,. Hence the two planes (54) and (55) are parallel to the z-axis, 
and their intersection is parallel to Oz unless these two planes coincide, that 
is, unless we have 


i (22) +2,(22) =. 


208 NON-LINEAR DIFFERENTIAL EQUATIONS | [Iv,§73 


Let us first discard the case in which this happens. The tangent to the 
curve I’ which passes through M, is parallel to Oz, but this curve itself does 
not present any singularity at the point M,. To assure ourselves of this, we 
shall replace the system of the two equations 


(57) i (¢,-¥, 2) ==.0; dy = zdz 
by the system of the two simultaneous equations 
(58) GEE EV ater 


— a= es a 
Oz 0z Ox oy 
with the initial conditions =z), y= Yp, 2 = Z%). The two systems are equiva- 


lent.’ In fact, from the equations (58) we derive the integrable combination 
dF = 0. Hence we have F(a, y, 2) = F'(%9, Yo, 2) = 9. Now, since 


oF oFf\. 
ears %0 e) 


does not vanish by hypothesis, we derive from the equations (58) the develop- 
ments of x — x, and of y— y) in powers of z— z, beginning with terms of at 
least the second degree, 


B= (2 —%)Pvy Yo = Bg(@— HIP ov 


The point M, is therefore an ordinary point for the curve I’ which passes 
through this point, but the projection m, of M, on the plane xOy is a cusp (in 
general of the first kind) for the curve c, the projection of I. This results, more- 
over, from a general property, which is easily verified, that the projection of a 
space curve on a plane, in a direction parallel to the tangent at a point M of 
the curve, has a cusp at the point m, the projection of M (I, Exercise 13, p. 582, 
2d ed.). If d denotes the projection of the curve D on the zy-plane, it follows 
that the curve d is the locus of the cusps of the integral curves c, as we have 
shown before. The preceding method has the advantage of showing us how this 
singularity disappears when we pass from the plane to the surface S. 

The result is quite different when the relation (56) is satisfied at all the 
points of the curve D. The two planes (54) and (55) are then coincident, and 
we have the case in which there exists a singular integral. Through every point 
of D there pass in general two curves I, the curve D itself and the second curve 
whose projection on the zy-plane is tangent to the singular integral curve d. 


74. Singular integrals of systems of differential equations. The theory of the 
singular integrals may be extended to systems of differential equations of the 
first order, and therefore also to equations of higher order. We shall study 
only a system of two equations of the first order (which covers also the case of 
a single equation of the second order), and we shall employ a process which is 
the reverse of the preceding —that is, we shall consider first of all a system 
obtained by the elimination of the constants.* Let ° 


(59) F(x, y, z; a,b) =09, (x, y, Z; a, b+) =0 


* See E. Goursat, Sur les solutions singulieres des équations différentielles 
simultanées (American Journal of Mathematics, Vol. XI). 


IV, § 74] SINGULAR INTEGRALS : 209 


be the equations of a family of plane or skew curves which depend upon two 
arbitrary parameters a and 6. Such a family is called a congruence of curves. 
Let us suppose, for simplicity, that the functions F and # are polynomials. The 
eurves of the congruence are then algebraic. We shall first generalize the 
theorems established for the congruences of straight lines (I, § 255). If we 
establish a relation between a and 6 of arbitrary form b = ¢(a), we obtain an 
infinite number of curves I depending upon a single arbitrary parameter a. 
In general these curves do not have an envelope. In order that an cnvelope 
exist, it is necessary that the four equations (59) and (60) shall have a systcin 
of common solutions in a, y, 2 (I, § 215, 2d ed.; § 223, 1st ed.) : 


(60) es cs Le oF OR eG 
ca ~=—s eb dda 6a ~—s 6b da 

The elimination of z, y, z from these four equations leads to a relation 

between a, b, and db/da, 
db 

(61) I («, b, x) a 
that is, to a differential equation of the first order. If we have taken for 
b = ¢(a) an integral of this equation, the curves I will generate a surface = 
and will be tangent to acurve C lying on 2. We shall call this curve C the edge 
of regression of X, as in the case of line congruences. If the equation (61) is of 
degree m in db/da, every curve I of the congruence belongs, in general, to m 
surfaces similar to 2, and it touches the corresponding edge of regression on 
each of these surfaces in a definite point. Thus there exist m remarkable par- 
ticular points on each curve I of the congruence, which we call the focal points. 
These focal points can be obtained without integrating the differential equa- 
tion (61), for we need only solve the four equations (59) and (60) for a, y, z, 
db/da. We find first the relation (61), which gives db/da, and, eliminating dh/da 
from the two equations (60), we have a new relation, 


(2) D(a,b)  éa a ab perry 
which, together with the two equations (59) of the curve I, enable us to calcu- 
late the coédrdinates of the focal points. 

The locus of the focal points is the focal surface of the congruence. We 
obtain the equation of this surface by eliminating a and b from the three rela- 
tions (59) and (62). The focal surface is also the locus of the edges of regres- 
sion C of the surfaces =. In fact, any point of the curve C is a focal point 
for the curve of the congruence which is tangent to C at that point. It follows 
that every curve I’ of the congruence is tangent to m sheets of the focal surface 
at the m corresponding focal points, since at each of these points it is tangent 
to a curve C lying on the focal surface. All these properties are exactly analo- 
gous to the properties of congruences of straight lines. In general, if F and ® 
are any polynomials, the m sheets of the focal surface are represented by a 
single equation, but it may also happen that this equation breaks up into sey- 
eral distinct equations. In certain particular cases it may also happen that 
some of the sheets of the focal surface reduce to curves. In such a case the 
corresponding edge of regression C reduces to a point. 


210 NON-LINEAR DIFFERENTIAL EQUATIONS [iv,§74 


The conclusion which we can derive from these properties with respect to 
differential equations is as follows: The curves T are the integral curves of a 
system of differential equations which is obtained by eliminating the constants 
a and b from the equations (59) and the equations obtained by differentiating 
them : 


63 a ey at re —+—y pops H 
Ue Be eey Paes i ees sy ees 

Let 
(64). S (25 Ys't, U5 2) 0, SCAT AOA SR 


be the system of differential equations thus obtained. The equations (59) rep- 
resent the general integral of this system, since by hypothesis we can choose the 
constants a and b in such a way that the curve I’ passes through any point 
(Xo Vo, Zo) Of space. If through this point there pass n curves I’, the equations (59) 
determine n systems of values for a and b. The equations (63) determine 
y’ and z’, and we see that for the point (9, Y¥9, Zp) the equations (64) determine 
n systems of values for y’ and z’. But the edges of regression C are also integral 
curves of the equations (64), since in a point of C the values of ag, y, z, y’, z 
are the same for C and for the curve I tangent to C at that point. The equa- 
tions (64) have, therefore, besides the integrals represented by curves I’, an 
infinite number of other integrals, not included in the equations (59), which 
are obtained by integrating the equation of the first order (61) ; these are the 
singular integrals of the system. 

On closer examination we see that the existence of the focal surfaces does 
not in reality require that the curves I shall be algebraic. It is sufficient that, 
in the neighborhood of a system of solutions (2%), Yo, Z%; @, 09) Of the three 
equations 

(65) F(a, y, Z, a, b) = 0, B(x, y, z, a, b)= 0, ee a as 
the implicit functions x, y, z of the parameters a and b, defined by these three 
equations, which reduce to 2, Yo, %, for a= a), b= 69, shall be continuous 
and have continuous derivatives in the neighborhood. In fact, let 


(66) x =f; (a, b), y =Se (a, 5), & = fs (a, b) 


be these three functions. The sheet of the focal surface which passes through 
the point (Zp, Yo, %) is represented in the neighborhood of this point by the 
equations (66), where the values of the parameters a and 6 are near a, and bo. 
It is easy to derive from this the equation of the plane tangent to the focal sur- 
face. In fact, when the point x, y, z describes any curve on this surface, a, y, 
z, a, b are functions of a single independent variable which satisfy the equa- 
tions (65) ; hence the differentials of these functions satisfy the two relations 


F 
og Ey + ee z+ sa + ab = 0, 
nj a or Oe 


Making use of the last of the relations (65), we can eliminate da and 6b, and 
‘we find the new relation 


D(F, ®) ,, , D(F, ®) 5, D(F, ®) 


D(q, b) Diy, b) Dé, Eats 


(67) 


IV, § 74] SINGULAR INTEGRALS 211 


We have only to replace dz, dy, 6z by X — 2%, Y — yo, Z— 2%, respectively, in 
order to have the equation of the plane tangent to the focal surface. It is easy 
to show that this plane passes through the tangent to the curve I. The prop- 
erties of the focal surface suppose, therefore, only that we can apply the 
theory of implicit functions to the equations (65), and in particular that the 
functions Ff, ®, together with their partial derivatives, are continuous in 
the neighborhood of a system of solutions %, Y¥9, 2; @, 09. This is certainly 
true when F and ® are polynomials, but it is clear that it is also true for 
many other functions. Let us also observe that if the curves I have singular 
points, the locus of these singular points forms a part of the focal surface. 
This is shown as in the case of the analogous proposition relative to plane 
curves (I, § 207, 2d ed.; § 201, Ist ed.). 

Let us now examine the question from the opposite point of view. Given a 
system of two differential equations of the first order, such as the system (64), 
let us propose to determine whether this system has singular integrals. We 
shall suppose that and SF; are polynomials. Let M, be any point (29, Yo, 2) 
of space. If x, y, z are replaced by 2p, Yp, 2, respectively, in the equations (64), 
these equations have in general a certain number of systems of solutions. Let 
Yo, % be one of these systems. Let us assume first that, for this system of solu- 
tions, the Jacobian D(a, A,)/D(y’, 2’) is not zero. From the equations (64), y’ 
and z’ can be found as regular functions in the neighborhood of the point 
(Xo, Yor Zo) 

Y =Yy + a(@— 2%) +++, z= % + ay (G— MH) +--+, 

which reduce to yp and z, respectively, fort=a), y=Yo, Z=2%. The equations (64) 
have therefore an integral curve passing through the point M, tangent to the 
straight line whose equations are Y— y, = yj (X — %), Z— % = %(X — %). 
Moreover, this curve forms part of a family of integral curves depending upon 
two arbitrary parameters (§ 26). This conclusion does not hold if we have 
D(P, A)/D (yg, %) = 9; but this can occur only if the codrdinates (x9, Yo, Zo) 
satisfy the relation 


(68) RAL, U2) = ©, 
which is obtained by eliminating y’ and z’ from the three equations 
D(F, S;) 
69 F = 0 Fo 0 ee I), 
( ) oS ’ Sy; ’ Dy’, z’) 


The equation (68) represents a surface S, and, from what we have just seen, 
an integral curve which does not lie on the surface S cannot be a singular 
integral curve. 

If the point M, is on the surface S, the three equations (69) have for this point a 
system of common solutions, y’ = yj, 2’ = Z. If the straight line D represented 
by the equations 

(70) pe a la Siete TLR 

1 Yo %9 
is not tangent to S (which is the general case), there is an integral curve pass- 
ing through the point M, and tangent to the straight line D. It has been shown 
that the point M, is in general a cusp for that curve. What is essential for us 
is that this integral curve cannot be on the surface, since its tangent is not in 
the tangent plane. In order that singular integrals may exist, in each point of 


212 NON-LINEAR DIFFERENTIAL EQUATIONS  [iv,§ 74 


S the corresponding straight line D must therefore be situated in the plane 
tangent to the surface. This condition is sufficient, for then through each point 
of S there passes a curve lying on the surface and tangent to the line D. These 
curves are determined by a differential equation of the first order, and they are 
indeed singular integral curves, for at each of their points the values of y’ and 
of z’ form a multiple system of solutions of the equations (64). 

Example 1. Consider the simultaneous system of equations 

(71) y—ay=0, w22=277+7?-1. 
The two values of 2’ are equal for all the points of the cylinder 2? + y2— 1=0, 
and the direction corresponding to that double roct is the perpendicular dropped 
from the point (z, y) on the z-axis. Since this perpendicular is not in the tangent 
plane to the cylinder, there cannot be any singular integrals. In this example it 
is easy to verify that the cylinder is the locus of the cusps of the integral curves, 
for the general integral of the system (71) is represented by the equations 


y= Cy, z=Va? + y? —1— are tan-V2? + y2? —14 oe 
Example 2. Every system of differential equations of the form 
(72) F(y—ay’,2—22,y,2)=0, PB(y—«y’, 2-22’, y, 2) =), 


which may be considered as a generaiization of Clairaut’s equation, is easily 
integrated by observing that the preceding relations lead to the equations 


( ~) sh & =) - 
ee ee (ne eel a ee, 
oy’ ou Oz’ ov 


EP yy. =) ”. (= =) ” 
x —)y”’ + — 2 —)2z’=0, 
ae ou Oz’ ov 


where u=y—2y’,v=z— 22’. These last equations are satisfied by assuming 
that y” = 0, 2” = 0, or by supposing that we have 


oF oF \ (0® of ofr oF \ /o® O® 
(73) — 2 — x — — © — —x—)=0. 
oy’ ou | \ dz’ ov 02’ ov / \oy’ ou 


Under the first supposition, y’ and 2’ are constants a and 6; whence we see 
that the curves which correspond to the general integral are he straight lines 
of the congruence represented by the two equations 


F(y — az, z— baz, a, b) = 0, &(y — ax, z — br, a, b) = 0. 


There are also singular integrals, since the straight lines of the congruence 
are tangent.to the two sheets of a focal surface. These singular integrals are 
the edges of regression of the developables of the congruence, and are obtained 
by the integration of a differential equation of the first order. The equation of 
the focal surface is obtained by eliminating y’ and. z’ from the relations (72) 
and (78). 


EXERCISES 
1. Examine the following differential equations for singular solutions : 
y/2 + (z nm =) y’ — (14+ 2) y— a = 0. . [SERRET.] 
ry*y? — yey’ + ata = 0. [ScHL6mixcu. | 
y?—2QaVyy + 4yVy=0. [Boote. | 
(vy — y)? — 2ay (1+) =0. toe 


2ay(1+ y?)— (zy + y)? = 9, [Mozeno.] 


IV, Exs. ] EXERCISES 213 


2*. The equation H(z, y) = 0, obtained by eliminating y’ between the two 
relations F(a, y, y’) = 0, oF/dx+ éF/dyy = 0, represents the locus of the 
points of inflection of the integral curves. 

Deduce from this the theorem of § 72, in regard to the locus of the cusps of 
the integral curves, by means of a transformation of reciprocal polars. 

[DarBoux, Bulletin des Sciences mathématiques, Vol. IV, 1878.] 


3. Determine the singular integrals of the system of differential equations 
y=’ +y24 2, SN Hee i [SERRET. ] 


4, Determine whether the differential equation of the second order, 
2 
ALES Ed Bisel (2 ay’ + 5M + yy? + ay’ —y=0, 


has singular integrals, and find any that exist. [LAGRANGE. ] 
[Replace this equation by a system of two equations of the first order.] 


5*, Given a differential equation of the second order, 
F(x, YwY,Y)=9, 
by eliminating y” between this equation and the relation 6F/dy” = 0 we obtain 
a differential equation of the first order P(z, y, y’) = 0, whose integrals have 
in general the following property : Through each point M of one of these inte- 
gral curves C there passes an integral curve of the equation F = 0, which has a 


cusp of the second kind at M, and whose cuspidal tangent is the tangent to the 
curve C at this point. [American Journal of Mathematics, Vol. XI, p. 364.] 


6. Establish the properties of ex by starting with the general integral of the 
differential equation dz/x + dy/y = 0, written in the algebraic form ry = C. 

Consider the same question for the function tana, finding first the general 
integral in algebraic form of the differential equation 


dx a Ee di 
l+a? 1+ y— 
7*, Let y = R(a, y), where F(a, y) is a rational function of y whose coeffi- 


cients are analytic functions of z, be a differential equation of the first order 
having a general integral of the form 


zt) yn rz) yn +... nl Z 
(1) Pol yy + $4( yy = FP) Hoe: y) =O. 
Yo(e) y" + Yy(H)y"—1 + e+ + Yn(Z) 
Prove that this equation can be reduced to a Riccati equation by a substitution 
of the form u = R,(z, y), where R,.is a rational function of y. [PAINLEVE.] 


Note. It will be noticed that the equation (1) can be written in the form 
y" + [Ay(2) + By (a) w]y"—2 + +++ + [An—a() + Br—a@) uly + u = 0, 
where u = (¢n — Cn)/($9 — Cy); and that u satisfies a Riccati equation, while 
the functions 4;, B; are known. 
8. If we seek to determine the function f(a) so that the envelope of the 
straight lines z cos a + ysin a = f(a) shall be a given curve C, we are led to 
a differential equation whose general integral is represented by the straight 


lines which pass through a fixed point of C. The true solution is furnished by 
the singular integral. a 


CHAPTER V 
PARTIAL DIFFERENTIAL EQUATIONS OF THE FIRST ORDER ~ 


This chapter is devoted to the theory of partial differential equa- 
tions of the first order. We shall consider for the most part the 
reduction of the integration of an equation of this type to that of a 
system of ordinary differential equations. Although this reduction is 
not, in many cases, of any practical utility, it nevertheless possesses 
great theoretical interest, for it enables us to determine just how 
difficult the problem is. Although not all the arguments require 
that the integrals considered shall be analytic, we shall restrict our- 
selves to that case unless the contrary is particularly stated. 


I. LINEAR EQUATIONS OF THE FIRST ORDER 


75. General method. We have already seen that the integration 
of the homogeneous equation 


x, 43,2 af _ 
(1) wt hagy to tage = 
where X5X,,-+>, X,are petal of x,, #,,+++, ,, and the integra- 
tion of the svatets of differential equations 
d 
@) Tape ha aol 
1 rea n 


are equivalent problems (§ 31). If fi, fy -++) fr-1 are (n — 1) inde- 
pendent first integrals of the system (2), the general integral of the 
equation (1) is an arbitrary function, 


O(f, ifs oe elas 


of these (n — 1) integrals. 

We can obtain the integral satisfying the Cauchy condition as 
follows: Suppose that the coefficients X; are analytic in the neighbor- 
hood of a particular system of values x%, x8, ---, #®, and that the first 
coefficient (X,), does not vanish at that point. Solving equation (1) 
with respect to éf/éx,, we can apply to it the general theorem of 
§ 25. Hence there exists an analytic integral in the neighborhood 
mentioned, which reduces, for x, = x?, to a given analytic function 

214 


V,§75] LINEAR EQUATIONS OF THE FIRST ORDER 215 


(x,, %, +++, ,) of the (n —1) variables x, x,,---, «,. In order 
to obtain this integral, let us write the system (2) in the form 
di, Xz el ew 


(°) dz, X,’ ies Cai bact 

where the right-hand sides are analytic in the neighborhood of the 
point (at, xg,---, #2). There exists a system of analytic integrals 
reducing to given values C,, C,,-+--, C, for .#, =, provided that 
each of the absolute values |C; — a?| is less than a certain limit, and 
these integrals are analytic functions of x, and of the parameters C,, 


Cy, +++, C, (§ 26), which are represented by developments of the form 
(4) x; = C; + (a 74; xy) P; (a, Ca, C’gy ++, C,). (@ = 2,3,+> i) 1) 
Solving these (n — 1) equations for the Cs, we obtain a system of 


(n —1) first integrals of the equations (2), ei dsarares by the 
developments 

(5) C;= 2@; + (a, me 18) Q(t, Hay**% *y Xn), (@ = 2, 3,-+:, 7) 
where the Q,’s are analytic functions. It is clear that the function 
$(C,, Cy,-++, Cr) of these (n —1) first integrals is analytic in the 
neighborhood of the point (a}, ---,a,) and reduces to  (a,, X,, +++, X,) 
POE cee oye 

Let us now consider any linear equation 


Oz 0 0 
(6) Pema eras — 0, 
1 n 


where P,, P,,-+-, P,, R may depend both upon the independent 
variables x,, #,,-+--, x, and upon the dependent variable z. We 
shall reduce this equation to the form (1) by means of a device 
very often used in the study of partial differential equations. 
Instead of trying to find the unknown function z directly, we shall 
try to define it by means of an equation not solved for z, 
(7) V (%, Ly, Ly +++) Ly) = O, 

where the function V of the (n+1) variables z, x, x,,+-+, x, is 
now the unknown function. From this relation we derive, by 
differentiation, 

RAMEE opel, ei da A Lc 

Brey Oe Oar is é Ox Ox Ox! 
and, replacing @z/dx,,---, @z/éx, by the values derived from the 
preceding relations, the equation (6) becomes 


é 
(8) F(v) =P, +. PB +R a= 0. 
1 


0; 


216 PARTIAL DIFFERENTIAL EQUATIONS [V, § 75 


The new equation is of the form (1), and its integration is equiva- 
lent, to that of the system 


(9) ete 


hence we may state the following proposition : 


Tf Uy) Ugy +++) Up, aren independent first integrals of the system (9), 


every function z of the n variables x,,%,,+++, L,, defined by a relation 
of the form 
(10) B(U,, Uy, +++) Up) =O 


where ® indicates an arbitrary function of U,, Uy, +++) Uy WS aN 
integral of the equation (6). 


We cannot conclude from this that we obtain all the integrals of 
the equation (6) in this way. In fact, in order that the implicit 
function defined by the relation (7) be an integral, it is not neces- 
sary that we have identically F(V)=0; it is sufficient that the 
equation F( V) = 0 be a consequence of the equation V = 0. If, for 
example, we take for V an integral of an equation of the form 
F(V)=KYV, where K indicates a constant different from zero, the re- 
lation V = 0 still defines an integral of the equation (6). It is quite 
in order, therefore, to determine. whether or not the relation (10) 
gives all the integrals of the given equation. In order to prove 
that this is really the case, with certain exceptions which we shall 
state, let us suppose that in the n functions w,, w,, +--+, u, we replace 
z by an integral of the equation (6). The Sealine ee, are 
nm functions U,, U,,---, U, of the m variables ~,, x,, -- . If we 
prove that the ‘J acbih of these n functions is idoaeale ae it 
will follow that we have a relation of the form 


Y(U,, U,, eee bes = 0, 


and consequently that the integral considered satisfies a relation of 
the form (10) in which the function © is foes by i This Jaco- 
bian is of the form - 


Cu, Ou, Ou, © 04u, Ou, Cu, 

te, 1 Ps Ge: : bn, Pa Oe Bag Ps Bel wrio oo 
A= (2. 7) 

OU, OUn OUn CU, ‘ 

Be, | PEGE on, TP" Oy 


V,§75] LINEAR EQUATIONS OF THE FIRST ORDER 217 


Noting that certain determinants in the development of A have two 
columns identical and therefore vanish, we may write 


= D(a, Ug, ***%, Un ~— D(w, Ug, ** "5 Un) 
fide LS se er reer 


D (yy gy sy By Free | (Hyp ty pia, By egy, + + *y-Ly) 


But, since w,, u,,++-, u, are nm first integrals of the system (9), we 
have 


“ Cu; Ou; 


my dies ie 0x, 


Ou; Ou, 
~ v + R v = 
OX 0z 


ane Cpr 0; (=1,2,.-.,n) 


hence, by the theory of linear homogeneous equations, we have 
R —. P; 
(12) 
D (Uy, Ug, > ++, Un) D(Uy, Ug) +++) Uy) 
D(x,, Ho * *) Xn) D(x,, Puce ghee — 19 eshte La ats Xn) 


(2 a lt2, ie, 1) 


= M, 


where M is a function of «,, x,,---, %,, # which we can always cal- 
culate when we know the first integrals w,, u,,---, u,. Substituting 
in (11) the values of the determinants deduced from (12), we find 


(12') MA = R — P.p,— P,p,— +++ — PrP 


If z is an integral of the equation (6), the right-hand side is zero; 
hence this integral satisfies either the condition A = 0 or else M = 0. 
In the first case, as we have just shown, this integral is defined by 
a relation of the form (10). As for the relation M = 0, it can define 
only one or more completely determined implicit functions. Hence, 
except for certain exceptional integrals which do not depend upon 
any arbitrary constant, all the integrals of the equation (6) satisfy a 
relation of the form (10). We shall hereafter say that the relation (10) 
represents the general integral of the equation (6). 


To see if an integral can satisfy the relation M = 0, let us consider any point 
of that integral, (x{, 7$,---, 72, z)), and let us suppose that all the coefficients 
P,, P.,+++, Pn, R are analytic in the neighborhood of this system of values 
without being all zero simultaneously for 7;= 2), z=z,. Let us assume, for 
example, that P, is not zero for this system of values. We can then solve the 
equation (8) for dV /dz,, and, by Cauchy’s theorems (§ 25), we can take for u,, 
Uy, +++, Un functions analytic in the neighborhood of this system of values. Now 


one of the equations (12) can be written in the form 
— py = MP (ty tay to), 
DZ, Loy ry Ln) 


Since the determinant on the right is analytic, and since P, is not zero for 
a, = x), Z = Z, it follows that this system of values cannot make M zero. Since 


218 PARTIAL DIFFERENTIAL EQUATIONS LV, § 75 


the point (x{,---, x2, 2) is any point of the integral, we see that there cannot 
exist integrals satisfying the relation M = 0 except in the two following cases : 
1) There exists a function V (2, %,+++, %,, 2) such that every system of 


values of the variables 2;, z that makes the function V vanish, also causes P,, 
P,,-++, Pn, and R to vanish. All these coefficients are therefore divisible by 
the same factor, and it is clear that by equating this factor to zero we obtain 
an integral. This trivial case is of slight interest. 

2) The reasoning would again be faulty if the integral defined by the rela- 
tion V = 0 were such that, in the neighborhood of every system of values 
satisfying that relation, some of the coefficients P;, R ceased to be analytic. 
This case can actually occur, as we shall show presently. ~ 


76. Geometric interpretation. The preceding general method is 
susceptible of a simple geometric interpretation in the case of an 
equation in three variables, which we shall write in the customary 


notation, 


0 0 
(18) Pp + aq =R, ina ae! qg=5 


x oy i 
where P, Q, RF are functions of the three variables x, y, x Let S be 
any integral surface. Since the equation of the plane tangent to this 


surface is 
Z—z=p(X—«“)+q(Y—y), 


the relation (13) expresses the fact that this tangent plane passes 
through the straight line D represented by the equations 


(14) a an eee ee 


Hence the problem of the integration of the equation (13) may be 
stated in geometric language as follows: 


To each point M of space, whose codrdinates are (x, y, 2), there 
corresponds a straight line D through that point, represented by the 
equations (14). A surface S is to be determined so that the tangent 
plane at each of its points passes through the straight line associated 
with that point. 


The surfaces possessing this property constitute the general inte- 
gral of the linear equation (13). The three functions P, Q, R deter- 
mine the law according to which the straight line D moves when the 
point M changes its position. These three functions are usually 
analytic functions of x, y, 2, but it is sufficient for the argument 
that they satisfy the conditions stated in our previous study of 
differential equations ($§ 27 ff.). 


V,§76] LINEAR EQUATIONS OF THE FIRST ORDER 219 


The preceding statement leads us to seek the curves I which are 
in each of their points tangent to the corresponding straight line D. 
We shall call these the characteristic curves. We shall first show 
that every integral surface is generated by characteristic curves. 
Consider, in fact, such a surface S. In each point M of that surface 
the corresponding straight line D lies in the tangent plane. We can 
therefore propose to determine the curves on that surface which are 
tangent at each of their points to the corresponding straight line D. 
These curves may be obtained by the integration of a differential 
equation of the first order ($17). Through each point of S there 
passes in general one and only one curve, possessing this property, 
which lies entirely on the surface. It is clear that these curves are 
characteristic curves, which proves the proposition. 

The converse is almost self-evident. If a surface is a locus of 
characteristic curves, the tangent plane at any one of its points con- 
tains the tangent to the characteristic curve lying upon the surface 
and passing through that point — that is, the straight line D. The 
given problem is therefore reduced to the determination of the 
characteristic curves. : 

The differential equations of these curves, by their very definition, 


are of the form 
dx dy dz 

() tia OUR 
Through each point of space there passes, therefore, in general one and 
only one characteristic curve tangent to the corresponding straight 
line D. Suppose that we have integrated these equations (15). 
Let w and v be two independent first integrals of this system. 
The general integral is represented by the equations 


(16) U(x, Y, 2) — a, U(X, Y, #) = 0, 


where a and 0 are two arbitrary constants. The characteristic curves, 
which depend upon two parameters, therefore form a congruence. In 
order to obtain a surface generated by the curves of this congruence, 
we must establish between the two parameters a and 6 an arbitrary 
relation, say @(a, 0) = 0, and the corresponding integral surface will 
have for its equation ¢(u, v)= 90. This is exactly the result to which 
the general method of the preceding paragraph would lead us, for u 
and v are here two independent integrals of the equation 

Ou Ou bu 


J Depedore apie’ 


7 tha, = 0. 


220 PARTIAL DIFFERENTIAL EQUATIONS [V, § 76 


Example 1. Consider the equation px + gy = mz. The differential equations 


of the characteristic curves, 
dx dy dz 


z y me’ 

have the two first integrals y/z = a, z/x™ = b, and the general equation of the 
integral surfaces is z = a™f(y/z). If m = 1, the characteristic curves are straight 
lines passing through the origin, and the integral surfaces are cones having their 
vertices at the origin. If m = 0, the characteristic curves are straight lines par- 
allel to the zy-plane and meeting the z-axis. The integral surfaces are conoids. 

Example 2. Consider the equation py —- gz+az=0. The differential equa- 
tions of the characteristic curves, 


Sd Ae 


give the two integrable combinations 
cdy —yde _ 0 


gy 


and the characteristic curves are represented by the equations 


edz + ydy = 0, dz— a 


? 


ai ya Cy, z—aare tan = C,. 

These are helices with the pitch 2 7a lying upon cylinders of revolution hay- 
ing Oz for axis, and the general integral is represented by helicoids (the axes of 
coérdinates being supposed rectangular). In the particular case where a = 0, 
the characteristic curves are circles having their centers on the z-axis and their 
planes parallel to the zy-plane. The integral surfaces are surfaces of revolution 
about the z-axis. 

Example 3. Orthogonal trajectories. Let 


(17) Ex y, 2) = 


be the equation of a family of surfaces 2 which depend upon an arbitrary 
parameter C in such a way that through every point of space (or at least of a 
portion of space) there passes one and only one of these surfaces. Let us con- 
sider the problem of finding another surface S, represented by the equation 


z= $(2, Y); 
which cuts orthogonally at each of its points the surface = through that point. 
Since the direction cosines of the normals to the two surfaces are respectively 


proportional to dF /dz, dF /édy, eF /éz for 2, and to p, g, — 1 for S, the condition 
of orthogonality leads to the linear equation 


(18) Dee a oO 


aoa dz 
(19) Sa a ae 

Cimon LOL 

Comey. L0f 


are the curves tangent at each of their points to the normal to the surface 3 
through that point. 


V,§76] LINEAR EQUATIONS OF THE FIRST ORDER 221 


Suppose, for example, that we have F(a, y, z) = 2f(x, y), where f(z, y) is a 
homogeneous function of the mth degree. The differential equations of the char- 
acteristic curves are here 

da _ dy _2dz 
IE eC We 
By Euler’s relation, we have the integrable combination 
zcdxz + ydy — mzdz = 0, 


from which we derive the first integral x? + y? — mz? =a. On the other hand, 
dy/dx is a homogeneous function of degree zero in the variables z, y. Hence 
we can obtain a new first integral by a quadrature (§ 3). 

Example 4. It is sometimes possible to determine the characteristic curves 
without any calculation, merely from their geometric definition. Let it be re- 
quired, for example, to determine the surfaces S such that the tangent plane at 
any point M of one of these surfaces meets a fixed straight line A in a point T, 
equally distant from the point M and from a fixed point O on the straight line A. 

Let M be a point in space; there exists on the straight line A one and only 
one point T such that TO = TM, and this point is the intersection of A with 
the plane perpendicular to the segment OM at its middle point. Let D be the 
straight line through the two points Mand T, The tangent plane to every sur- 
face satisfying the given condition and passing through the point M therefore 
contains this straight line D. Consequently these surfaces are obtained by the 
integration of a linear equation. Since the tangents to the characteristic curves 
all meet the straight line A, these curves are plane curves, lying in planes pass- 
ing through A. The characteristic curves lying in one of these planes are the 
integral curves of a differential equation of the first order, and it is easy to see, 
from their definition, that they are circles tangent to the straight line A at O. 
The required surfaces are therefore generated by the circles tangent at O to the 
straight line A. 


We can dispose of the arbitrary function $(w, v) in such a way 
that the integral surface passes through a given curve T'; we shall 
obtain that surface by taking the locus of the characteristic curves 
passing through the different points of the given curve. If I is 
represented by the system of two equations 

(20) O(a, y, 2) = 0, ® (x, y, %) = 9, 
the whole question reduces to finding the relation which must hold 
between the parameters a and 6 in order that a characteristic curve 
shall meet the curve I. It is clear that that relation may be found 
by eliminating a, y, between the equations (20) and the equations 
u = a, v = b of the characteristic curve. The problem has only one 
solution, unless the curve TI is itself a characteristic curve. In this 
singular case it suffices, in order to obtain an integral surface pass- 
ing through I, to consider the surface generated by a family of 
characteristic curves which depend upon an arbitrary parameter, and 
of which the curve I is a member. 


222 PARTIAL DIFFERENTIAL EQUATIONS [V, § 77 


77. Congruences of characteristic curves. To every linear equation 
of the form (13) there corresponds a congruence of characteristic curves 
formed by the characteristic curves of that equation. Conversely, 
every congruence of curves, that is, every family of curves depending 
upon two arbitrary parameters a and 4, is the congruence of charac- 
teristic curves for an equation of the form (13).* Suppose, in fact, 
that the equations which define that congruence are solved for the 
two parameters a and b: 

UAE, Yy'@) = th, v(a, Y, 2) = 0. 
Every surface S generated by the curves of this congruence, associated 
according to an arbitrary law, is represented by an equation of the 
form v=7(uw). Taking the partial derivatives with respect to x 
and to y, we find 
0 ou 0 é 

oe t ag = TOO Se + Ge) yt t= MOG + ed) 
The elimination of 7'(w) leads to a linear equation 


D(u, v) D(u, v) D(u, v) 
Dey? D@ 2) 0 Diy) a? 
for which the given congruence is evidently the congruence of 
characteristic curves. 
Let us now consider the general case of a congruence defined by 
two equations of any form whatever, 
(21) UF (yy 4.0, 0) 80, V (xj, 2, a, B=), 
If we set up an arbitrary relation ¢(a, }) =0 between the two 
parameters a and 6, we shall have the equation of a surface S gener- 
ated by the curves T of the congruence by eliminating a and 6 from 
the equations (21) and the relation ¢ = 0. All these surfaces again 
satisfy, whatever may -be the function 4, the same partial differen- 
tial equation of the first order. To obtain this equation we may 
proceed as follows: The three equations 


(22) Usd), vat); (a, 6) =0 
define three implicit functions z, a, b of the independent variables 
x and y, and the last contains only a and b. Hence we have 


* We suppose, in addition, that through any point of space (or of a portion of space) 
there passes one of these curves, which would not happen if they were all situated 
upon the same surface. . 


V,§77] LINEAR EQUATIONS OF THE FIRST ORDER 223 


On the other hand, if we differentiate the first two of the equa- 
tions (22) with respect to x and to y, we can derive from the result- 
ing relations expressions for @a/dx, 0b/0x, da/ey, eb/dy in terms of 
XL, Y, 2, P, Y, 4, 6, and, by replacing these derivatives in the determi- 
nant (23) by their values, we obtain a new relation, 


P(x, ¥, 2 P, J, a, 6) = 0. 


We need only eliminate a and 6 from this relation and the two rela- 
tions (21) in order to obtain an equation containing only a, y, z, p, q, 


(24) F(x, Y, 2% Pp, 7) = 9, 
which apples to all the surfaces generated by the curves of the 
congruence. It is easy to show, from the very way in which this 
equation has been obtained, that it breaks up into a system of linear 
equations in p and g. The same fact results from its meaning. Let 
us suppose, for definiteness, that through a point M of space there 
pass m curves of the congruence, and let D,, D,,---+, D,, be the m 
tangents to these curves at the point M. Every surface through the 
point M generated by the curves of the congruence must contain 
one of the m curves of this congruence which pass through M; 
hence the tangent plane at the point 17 must pass through one of 
the straight lines D,, D,,---, D,. Let P;, Q;,, R; be proportional 
to the direction cosines of the straight line D;. Every surface gen- 
erated by the curves of the congruence must therefore satisfy one 
of the m equations, 

(25) H,=Pyp+Qq—Rk,=9, 1 2 ey 20) 
and the left-hand side of the equation (24) is identical, except for a 
factor independent of p and of q, with the product of the m linear 
factors E,, E,,---, E,. It should be noticed also that it would be 
impossible, in general, to separate these m factors analytically. 

Similarly, certain problems of geometry may lead to partial differ- 
ential equations of the first order which decompose into a product of 
linear factors. Let us consider again, for example, the problem of 
the orthogonal trajectories to a family of surfaces whose equation 
F(a, y, 2, C) = 0 is of degree m in the arbitrary parameter C. To 
obtain the partial differential equation of orthogonal surfaces, we 
must again eliminate C between the relation F = 0 and the condition 


an | OF OF 

Pay * 2 Cy 02 

Through a point M of space there pass, by hypothesis, m surfaces 
of the given family. Let D,, D,,--+, D,, be the normals to these m 


224 PARTIAL DIFFERENTIAL EQUATIONS [V, § 77 


surfaces. The tangent plane to an orthogonal surface through M 
must contain one of these straight lhnes. Hence the partial differen- 
tial equation decomposes into a system of m equations which are 
linear in p and gq. 

Conversely, given any equation of this type, to each point of 
space there correspond m straight lines D,, D,,-+-, D,,, and the 
plane tangent to any integral surface contains one of these straight 
lines. If we give the name characteristic curve to every curve which, 
at each of its points, is tangent to one of the corresponding m straight 
lines, the reasoning employed above shows again that every integral 
surface is a locus of characteristic curves. To obtain the differential 
equations of these curves, we are not compelled to carry out the 
decomposition of the left-hand side of the equation into linear 
factors. Indeed, expressing the fact that the left-hand side is divisi- 
ble by the factor Pp + Qq7 — Rk, we obtain equations of condition 
homogeneous in P, Q, R, which furnish m systems of values for the 
ratios of these coefficients for each point (a, y, z). Replacing P, Q, 
R in these conditions by the proportional quantities dx, dy, dz, we 
obtain the differential equations of the characteristic curves, and 
the integration of the partial differential equation is reduced to the 
integration of a system of ordinary differential equations. 


The preceding theory explains very simply how a linear equation may have 
integrals which are not included in the general integral. Consider a partial 
differential equation of the form 


(26) EG, Vy 2,039) —=.0, 


whose left-hand side is the product of a certain number of linear factors in p 
and q that are not analytically distinct, and let 


dy dz 


27 P(x Zo Nee oe 
(27) (2, v2, 54 = 


)=0, ¥ (2, v, 2, =) =0 


dx’ da 

be the differential equations of the characteristic curves of this system. The 
curves which represent the general integral of this system form a congruence, 
which is the congruence of the characteristic curves of the equation (26), and 
the general integral is represented by the surfaces generated by the curves of 
this congruence associated according to an arbitrary law. But it may happen 
that the equations (27) have singular integrals. This will happen if the con- 
gruence of the characteristic curves has a focal surface (Z). Then through each 
point of this surface there passes a curve of the congruence of characteristics 
tangent to this surface. The plane tangent to (=) contains, therefore, one of 
the straight lines D; relative to the point of contact, and consequently (2) is 
an integral surface of the equation (26). Moreover, it is not a member, at least 
in general, of the surfaces which represent the general integral ; that is, it is a 
singular integral surface, 


V, § 78] TOTAL DIFFERENTIAL EQUATIONS 225 


Consider, for example, the equation 
(28) p (a? — 2) + q(@y t2V22+ 2-2) =0, 


which in reality is equivalent to two linear equations. We can write the 
differential equations of the characteristic curves in the form 


dz dy\? dy\? 
heyy Le CA 
dz (v on) | +(@) | 


The integration is immediate, and the congruence of characteristic curves is 
formed by the straight lines 


Z=-C, y=OC,7+2V14 C3, 


which are parallel to the zy-plane and tangent to the cone x? + y? = zg, The 
general integral is represented by the conoid surfaces generated by these straight 
lines, and there is a singular integral, the cone itself. 

The coefficient of g in the equation (28) is not analytic in the neighborhood 
of any point (%), Yo, 2) of this cone, which confirms a previous remark (§ 75). 


II. TOTAL DIFFERENTIAL EQUATIONS 


78. The equation dz = Adx+Bdy. The existence of integrals of a 
completely integrable system of total differential equations was estab- 
lished in § 24. The integration of such a system reduces to the 
integration of several systems of ordinary differential equations 
with a single independent variable. The method, which we shall 
develop only in the simplest case, is extensible to the general case. 

Let the equation be 


(29) dz=A (x, y, 2)dx+ B(a, y, 2)dy, 


where z is an unknown function of the two independent variables x 
and y. This equation is equivalent to two distinct relations 


02 Oz 
(30) ae =A (a, Y, ), a = B(a, y, 2). 


Every integral common to these two equations satisfies also the two 
new equations 
Gaines: CAL nC A C2 De 
Bedi se OMe Gee? Fie ee OEE, 


and consequently the relation 


OA dae) OB 
(31) — +> B=— 


0B 
Oy = Oz ioe Ox Oz 


451% 


If this relation does not reduce to an identity, there can be no in- 
tegrals of the given equation (29), except possibly one or more of the 
implicit functions defined by the equation (31). Hence in this case 


226 PARTIAL DIFFERENTIAL EQUATIONS [V, § 78 


we can always determine by substitution whether the equations (30) 
have a common integral. On the other hand, in order that these 
equations may have an infinite number of integrals depending upon 
an arbitrary constant, the relation (31) must be satisfied identically. 
If it is, the equation (29) is said to be completely integrable. 

In order to obtain all its integrals, let us first disregard the second 
of the equations (30), and consider only the first. If we regard y 
as a parameter, this equation is a differential equation of the first 
order between the independent variable x and the dependent vari- 
able 2; hence it has an infinite number of integrals z = (a, y, C) 
that depend upon an arbitrary constant C. We may replace this 
constant C by any function u(y) of the variable y, since the expres- 
sion for dz/éx remains the same when we replace C by a function 
of y. The solution of the problem therefore depends upon the deter- 
mination of this function w(vy) in such a way that.the derivative of 
the function z = ¢[a, y, w(y)] with respect to y shall be equal to 
B(x, y, @). This leads to the equation 


Od , Op du 
dy ay dy = Bla, y, (x, Y; u) |, 
or 
0 
a Bla, Y) (x, Y; ieee 
lin dance: er 


Ou 


We shall show that the right-hand side of this equation depends 
only upon the variables y and wu. It is sufficient to show that the 
derivative with respect to x is identically zero, that is, that we have 


(0B, OBOb OH) _ i 
oe les Bp Ob Ox. Ox =) Be, yb) — ole ox 


From the very manner in which the function ¢(a, y, w) has been 
obtained, we have the relation 


(34) st = A(x, y, >), 


which is satisfied for all values of x, y, and wu. It follows that we 
may write 

Ob =< 0A Op 

Be BU Wey, OVO 


—_— ——— 


V, § 78] TOTAL DIFFERENTIAL EQUATIONS 227 


Replacing 0/0x, ’p/0x dy, Ph/Ou dx by the preceding values, the 
relation to be verified reduces to the form 


26/08, OB, 24 94 ,\_, 
Ou\0x dO Cy 0d )= ; 


The second factor is identically zero by the condition of integra- 
bility (31). The equation (32) is therefore of the form 
du 
(35) eh) = F(y, u). 

Let u = W(y, C) be the general integral of this equation, where C is 
a constant independent both of # and of y. Then if we replace u by 
W(y, C) in the function ¢(a, y, vw), we obtain the general integral of 
the completely integrable equation (29), and we see that the integra- 
tion of this equation reduces to the successive integrations of two 
ordinary differential equations (34) and (35). 


Example. Consider the total differential equation 


ED) 


(36) das 
1+ cy 1+ xy 


dy, 


which is equivalent to the system 


oz 1+ y¥2 6z  “£(z—72) 


(36’) ie 5 ies 
ox «61+ xy oy 1+ zy 


The condition of integrability is verified, and the first of the equations (86’), 
which is linear in z and 0z/éa, has for its general integral 


z= T+ u(y) (1+ 29), 


where u(y) is an arbitrary function of y. Substituting this value of z in the 
second of the equations (36’), it becomes du/dy + 1/y? = 0, whence we derive 
u(y) =1/y + C. Hence the general integral of the equation (86) is 


(37) z=2+ C(1+ zy), 


where C indicates an arbitrary constant. 


The preceding problem can also be interpreted geometrically. 
In order to simplify the statement, we shall again call an integral 
surface any surface represented by an equation z = f(a, y), where 
the function f(a, y) is an integral of the equation (29). The two 
conditions (30), or 


p= Ae, Y, %); q= Ba, Y; z), 


228 PARTIAL DIFFERENTIAL EQUATIONS [V, § 78 


express the fact that the tangent plane to the integral surface S at 
a point (x, y, z) of that surface coincides with the plane P whose 
equation 1s 


(38) Z—2=A(X—2)+B(Y—y), 


so that the problem of the integration of the equation (29) is 
equivalent to the following geometric problem: 


To each point of space (x, y, #) there corresponds a plane P through 
that point, which is represented by the equation (38). It is required 
to find the surfaces S whose tangent plane at each point (x, y, 2) is the 
plane P associated with that point. 


The proposition is analogous to that of § 76. But in the present 
case the problem does not always have a solution. If the condition 
of integrability (31) is satisfied, there exists, in general, one and 
only one integral of the equation (29) which takes on a given value 
z, when x and y take on given values x, and y,. Through every 
point in space there passes, therefore, in general, one and only one 
integral surface. 

Let us consider, for example, a family of skew curves I’ which 
depend upon two arbitrary parameters a and 6, and which are rep- 
resented by a system of two equations 


(39) U(X, Y, 2) =a, Uk, Y; 2) = 0 

such that through every point of space (or of a region of space) 
there passes one and only one curve of this family. There does not 
always exist a family of surfaces S which has these curves I for 
orthogonal trajectories. In fact, the tangent plane to the surface S 
passing through a point would have to coincide with the normal 
plane to the curve I’ passing through the same point. We are there- 
fore led to a particular case of the preceding problem, which proves 
that the curves of an arbitrarily assigned congruence of curves are 
not, in general, the orthogonal trajectories of any family of surfaces. 
The plane tangent to the surface S through the point (a, y, z) must 
be perpendicular to the planes tangent to the two surfaces (39) 
which pass through the tangent to the curve I. Hence we have, in 
rectangular codrdinates, the two conditions 


aN, Ou yids IO © doy MIDRU DDE 
On? byl Ge? Oe 
From these equations the values of » and qg are found to be 


p=A(a,y, z), q = B(a, Ys z), 


V, § 79] TOTAL DIFFERENTIAL EQUATIONS 229 


and the condition (31) must be satisfied identically in order that the 
problem have a solution. 


Let us take, for example, the family of curves 
A OL, yo=t0L: 


where X is a function of z alone, and Y and Z are respectively functions of y 
alone and of z alone. The preceding method gives the following values for p 
and q, 

ee YZ’ 


Ss apsp ht nar ark 


and the total differential equation can be written in the form 


Lide Xx Ute ey Oy 
— 4+—— 4+ —=0. 

Te ne EX? ay 7 
It is clear that this equation is completely integrable, and the general integral 
is obtained by quadratures 


feet [eer [pune 


79. Mayer’s method. The preceding method requires two successive integra- 
tions. We can replace these two integrations by a single integration, as follows: 
Let us suppose, for definiteness, that the coefficients A(z, y, z) and B(z, y, z) 
are analytic in the neighborhood of the point (9, Yo, Z). Then there exists one 
and only one integral surface S, through the point (9, Yo, Zo) if the condition (31) 
is satisfied. Mayer’s method for obtaining this surface reduces to determin- 
ing first the sections cut from that surface by the planes parallel to the z-axis 
through the point (Xp, Yo, Z)). Let I’ be the intersection of S) with the plane 


(40) Y— Yo = M(x — X), 


where m has any given value. Along this curve I we have dy = mdz, and, replac- 
ing y and dy in the equation (29) by the preceding values, we obtain the relation 


(41) dz = {A[a, yy + m(e— a), 2] + mBle, yo + mw — ay), Z]} ax, 


which is also satisfied along the whole length of the curve I’. Now this is a 
relation containing only the two variables g and z; that is, it is a differential 
equation of the first order, the integration of which determines the curve I. Let 


(42) 2 O(Gis Loy Yo; 2651 1) 


be the integral of this equation which reduces to z, forz =a). The curve I is 
represented by the two equations (40) and (42). Since the required surface S, 
is the locus of the curves [' as the parameter m varies, the equation of this sur- 
face is obtained by eliminating m from the equations (40) and (42). To accom- 
plish this it is sufficient to replace m in the equation (42) by (y — ¥)/(@ — 2p). 
This method presents an evident analogy with the one which has been indicated 
for the integration of the total differentials P(x, y)dx + Q(a, y)dy (I, § 152). 
We might generalize it still further by replacing the planes parallel to the 
z-axis by cylinders passing through a given point (#), Yo, 2) and having their 
generators parallel to Oz. 


230 PARTIAL DIFFERENTIAL EQUATIONS [V, § 79 


For example, let us again take the equation (36), and let us suppose 2) = yy) = 0. 
Substituting y = mx, dy = mdz, that equation becomes 


dz  2%ma«z 1— mex? 
dx 1+ma2 > 14 mz? 


This is a linear equation which is readily integrated, and the integral which re- 
duces to Z) for x = 0 has the form 
zZ2=2+ z,(1+ mz’). 


Hence the surface S, has the equation z = £ + z)(1+ zy), which is the result 
obtained by the first method. 


80. The equation Pdx + Qdy + Rdz=0. The problem of the inte- 
eration of a total differential equation can be put in a more general 
and more symmetrical form. Let P(a, y, z), Q(x, y, 2), R(a, y, 2) be 
three functions of the variables x, y, z. To integrate the equation 

(43) P(x, y, z)dx + Q(x, y, 2)dy + R(a, y, z)dz= 0 
is to find a relation F(x, y, z)=0 between a, y, z such. that these 
three variables and their differentials dx, dy, dz satisfy the given rela- 
tion. If the function F contains the variable z, we may regard x and 


y in it as two independent variables and z as a function of these two 
variables, and we see that that function must satisfy the equation 


i Q 
dz =~ at — py, 
which is of the form (29). Replacing A by — P/R and B by — Q/R, 
and carrying out the differentiations, the condition of integrability (31) 
becomes 


CQ aman Chi OP OP shag 

(44) AG oy aah ee 7 ed 
This condition remains the same when we permute «, y, z and P, Q, R 
circularly. Hence we should have obtained the same relation if, in- 
stead of regarding z as the dependent variable, we had taken one of 
the variables x or y for the unknown dependent variable. The prob- 
lem of the integration of the equation (43), therefore, does not differ 
essentially from the problem already treated; but when we write 
a total differential equation in this way, it is not necessary to - 
specify which of the variables have been chosen as the independent 
variables. 

The condition (44) arises in a question which is closely connected 
with the preceding. Given an expression 


P(a, y)dx + Q@, y)dy, 


V, § 80] TOTAL DIFFERENTIAL EQUATIONS 231 


we have seen (§§ 12, 26) that there always exist an infinite number 
of factors w(a, y) such that the product w(Pdx + Qdy) is the total 
differential of a function of the two variables 2 and y. When we 
pass from two to three variables, this does not remain true in general. 
Let us consider, in fact, three functions, P, Q, R, of the variables 
x, y, 2. In order that the product w(Pdx + Qdy + Rdz) be an exact 
differential, the factor (a, y, z) must satisfy the three conditions 


OQ) OUR) (mR) _ OP) (MP) _ 8 (HQ) 
Oz oy x Oz oy 0x 


If we add these three equations, after having multiplied them by 
P, Q, R respectively, and then divide by yp, we find again the con- 
dition of integrability (44). This condition is therefore necessary 
in order that the trinomial Pdx + Qdy + Rdz have an integrating 
factor. It is also sufficient. For if it is satisfied, the equation (43) 
is completely integrable. Let 


(45) FENG, 2) 'C 


be the general integral of this equation. The values of ¢éz/éx and of 
éz/dy derived from the equation (45) must be identical with the 
values — P/R and — Q/R obtained from the equation (43), since we 
can choose the arbitrary constant C so that the integral surface 
passes through any point of space. For this we must have 


—. —_— — ot 
— 


or 
dF = u(Pdx + Qdy +Rdz). 


The factor w, which is equal to the common value of the preceding 
ratios, is therefore an integrating factor. Repeating the reasoning 
of § 12, we see, in a similar manner, that there are in this case an 
infinite number of integrating factors, which are of the form pwr(F), 
where 7 is an arbitrary function. 


The condition of integrability (44) is invariant with respect to every change 
of variables. Consider, in fact, a transformation defined by the equations 


(46) wt =f(u, v, Ww), ue ? (U, v, w), zZ=y(U, %, W), 
where the Jacobian of the functions f, ¢, y with respect to u, v, w is not identi- 
cally zero. This transformation carries the trinomial Pdz + Qdy + Rdz into an 


expression of the same form, P,du + Q,dv + R,dw, where P,, Q,, Rh, are func- 
tions of u, v, w. If now the relation (44) is satisfied, the analogous relation 


aQ, oR aR, oP aP, aQ 
47 Pp ah) (=o - =) R (3 - ay, 
G7) (= ov 4% ou ow ie ov ou 


232 PARTIAL DIFFERENTIAL EQUATIONS [V, § 80 


is also satisfied identically. We might verify this by a direct calculation 
(I, Chap. III, Ex. 19, 2d ed.; I, Chap. II, Ex. 19, Ist ed.), but it also results 
from the meaning of the condition. In fact, if the relation (44) is satisfied, 
there exist two functions u(x, y, z) and F(x, y, z) such that 


u(Pdz + Qdy + Rdz) = dF. 


If we carry out the change of variables defined by the equations (46), the 
functions u and F change into two functions y,(u, v, w), F,(u, v, w) of the new 
variables, and we have identically dF =dF,. Hence the preceding identity 


becomes 
B, (P, du + Q, dv + R, dw) = dF,, 


and the trinomial P, du + Q,dv + R,dw has an integrating factor. This shows 
that P,, Q,, R, satisfy also the relation (47). 

This remark enables us to present the method of integration of § 78 under a 
more general form. For let us suppose that the trinomial Pdz + Qdy + Rdz has 
been converted by a transformation into a binomial of the form P, du + Q, dv, 
containing now only two differentials du and dv. In the relation (47) we must 
suppose Rh, = 0, and that relation reduces to 


which shows that the ratio of the two coefficients P, and Q, is independent of 
w. The integration of the given total differential equation is therefore reduced 
to the integration of an equation of the form dv + mw (u, v)du = 0, that is, to an 
ordinary differential equation. 

Every trinomial Pdz + Qdy + Rdz can be reduced to a binomial P, du+ Q, dv 
in an infinite number of ways. For example, we can proceed as follows: We 
determine first two functions, w(x, y, z) and F(a, y, z), such that, whatever dz 
and dy may be, 

nls + his =p[P(a, y, z2)dx+ Q(z, y, 2) dy]. 

Ox oy 
This amounts in reality to integrating the differential equation Pdz + Qdy =0, 
regarding z as a parameter. Again, we may write the preceding equation in 
the form 


dF + (uk — ae = w(Pdz + Qdy + Rdz). 
Zz 


Then if we select a new system of independent variables, of which F(a, y, 2) 
and z are two, we see that Pdx + Qdy + Rdz is actually replaced by an expres- 
sion in which there appear only the two differentials dF and dz. This procedure 
can be varied in many ways. It is clear, for example, that we can begin by 
integrating either of the two equations 


Qdy + Rdz =0, Pdz+Rdz=0; 


this last method is in reality identical with the method of § 78. 
We can also connect with the preceding remark an elegant method due to 
Joseph Bertrand. Assuming that the equation (48) is completely integrable, 


V, § 80] TOTAL DIFFERENTIAL EQUATIONS 233 


let us begin by integrating the linear partial differential equation 

7) oR\é Often PaL\ oO CEIEOO\C 
Se eae (eae 

Oz oy / ox oy Oz 


Ox Oz 
Let u and v be two independent integrals of this equation. If between the two 
relations 


48 xy =( 

(48) (/) 7 ge 
X (u) = 0, An(0) 0 

and the condition of integrability (44) we eliminate the three differences 


2Q oR aR oP eP aQ 


0z oy : Ox ez” oy Ox ‘ 


we obtain the equality 


du dU OU 
Co. Oy 20z 
dv ov av/=0. 
COL 0 ae 
Fete he: 
There exist, therefore, two functions \ and yw for which we have 
ou Ov ou Ov ou Ov 
(49) P=\—+4 Q=A—+ 4—> R=rAj—+p—, 
Ox Ox oy oy Oz Oz 


and we can write the given equation in the form 
Adu + wpdv = 0. 


Now we have seen that the ratio \/u can depend only upon the variables u and 
v; hence this equation is a differential equation in u and v. 

This method appears to be more complicated than the preceding, since the 
integration of the equation (48) requires first the integration of a system of two 
differential equations of the first order. But it is more symmetric, and it may 
be preferable if the given equation is itself symmetric in z, y, and z. 

Consider, for example, the equation 


(y? + yz + 27) da + (22 + zx + 2?) dy + (x2 + zy + y”)dz=0. 


The condition (44) is satisfied, and the linear equation (48) is here 
of of of 
z2—y)— + (&— 2) — —2)—=0. 
(oe iil eet rar ae es 


The corresponding system of differential equations, 


ite ie Oe 
2 ye 2 x 


’ 


gives easily the two integrable combinations 
d(jtc+y+z)=0, rdz + ydy + zdz=0. 


Hence we may take 

u=r+yt+Z, v= a? + y? + 2, 
and the values of the factors \ and uw derived from the equations (49) are 
ut + v 5 etyte. u 


’ .=— ——___- =- -. 


2 2 2 


A= e+ yP? + 22+ ay + yz+ w= 


234 PARTIAL DIFFERENTIAL EQUATIONS [V, § 80 


The transformed equation in u and v is therefore 
(u? + v)du — udv = 0, 


y2 


or 


It follows that the general integral is u— v/u = C, or, returning to the vari- 


ables 2, y, 2, ay + yz + 2a 


ohyte 


81. The parenthesis (u, v) and the bracket [u, v]. Any total differ- 
ential equation is really equivalent to two simultaneous equations 


pHAaY, %), q= Ba, Y; 2). 
Let us now consider any two equations, 


(50) F(a, Y, *) P q) = 0, d (a, Y; * Py q) = 0 
in the two independent variables x and y, the unknown dependent 
function z, and its two partial derivatives p and q. 

If we can solve these two equations for p and g, we obtain two 
equations, p = f(a, y, 2), g = $(&, y, 2), of a form which has already 
been studied, and it will be possible to determine whether these two 
relations are compatible. But we can determine whether the condi- 
tion of integrability is satisfied without first solving the equations 
(50) for p and g. We have only to apply the rules for the calcula- 
tion of the derivatives of implicit functions. Let us consider, in fact, 
the relations (50) as defining two implicit functions, p = f(a, y, 2), 
gq = $(a, y, #), of the three independent variables a, y, z. Differen- 
tiating with respect to x, we find 


OF 7, OF op OF Oq _ Lea ice ony, Boag 
Ox «Op On Og Ox” OL; Op Ot Clog nor 


and consequently 


= 0, 


D(F,®) 8g | D(F,®) _ 
D(p,q) 0x  D(p, x) 
Similarly, we have 
D(F,®)ép . D(F,® D(F, &) @ F,®) _ 
D(p,9) ¢4y DY, 9) D(p,q) ex De 7) 
D(F,®) 8g | D(F,®) _ 
D(p,q) 0 D(p,#) 
Substituting the values of 0p/0y, Op/éz, 0q/0x, 0q/ez in the con- 
dition of integrability 


oy a ~ Oa Oz ?? 


V, $81] TOTAL DIFFERENTIAL EQUATIONS 235 
that condition becomes, after development, 


On (ae, 20) OF (e@ , ae 
Op \ ex ae dq \ ey te) 


0® /0F OF 0® (OF OF 
— ap las +? Ge) ~ By (ay 12%) 


In general, if w and v are any functions of x, y, z, p, g, we shall set 


Cat Oe AC ge 
de dx ' © dz’ dy oy 1 oz 
Oudv dvdu dudv dvdu 


—S i —— eee eee 


and we shall call the expression [u, v] a bracket. The preceding 
condition can then be written in an abridged form, 


(51) [F, ®]=0. 


In order that the two equations (50) shall form a completely inte- 
grable system, it must first be possible to solve them for p and q; 
that is, it must not be possible to derive from them a relation 
between a, y, independent of p and of q; and, further, the con- 
dition [F, &] = 0 must be a consequence of the two relations (50). 
If the bracket [F, ©] is identically zero, the two equations F = a, 
® =} form a completely integrable system for any values of the 
constants a and d. If the relation [F, &] = 0 is a consequence of the 
single equation F = 0, independently of the second equation & = 0, 
the two equations F = 0, ®= ) form a completely integrable system 
for any value of the constant 0. 

If the two functions F and @ do not contain z, the expression for 
the bracket [F, ®] is simplified. The following expression, where u 
and v are any functions of a, y, p, q, 


is called the parenthesis (u, v). The condition that the two equations 
F(a, YP» q) = 0, ® (x, Y; P) q)=9 
be compatible is, by what precedes, that the equation 
(F . ®) = 0 


shall be satisfied, either identically or as a consequence of the 
relations F= 0 and = 0 themselves. 


236 PARTIAL DIFFERENTIAL EQUATIONS [V, § 82 


III. EQUATIONS OF THE FIRST ORDER IN THREE 
VARIABLES 


82. Complete integrals. We shall now consider the integration of 
a partial differential equation of the first order, of any form what- 
ever but with only two independent variables, and we shall first 
present some very important results obtained by Lagrange. Let 


(52) F(a, y, % py 7) =9 
be the given equation. The fundamental result obtained by Lagrange 
is the following: If we know a family of integrals which depend 
upon two arbitrary parameters, we can derive all the other integrals 
from them by differentiations and eliminations. Let 

(53) Vids U2, 0,0) =O 
be a relation which contains two arbitrary constants a and 6, and 
which defines an integral of the equation (52) for any values of 
those constants. The values of the partial derivatives p and q of 
that integral are given by the equations 


OV OV OV 0V 

(54) Sais Dane gin a? ra ep 
By hypothesis, the function z always satisfies the equation (52) for 
any values of a and é; hence the elimination of the two parame- 
ters a and 6 from the three relations (53) and (54) will lead to the 
equation (52) and to that one only.* 

We shall now show that this equation (52) expresses the neces- 
sary and sufficient condition that the three equations (53) and (54) 
be satisfied by a system of three functions z, a, b of the two varia- 
bles x and y, where p and qg denote the partial derivatives of z with 
respect to x and y respectively. When this has been proved, it will 
be evident that the problem of integrating the single equation (52) 
is equivalent to the following problem: To find three functions z, a, 
b of the two independent variables x and y which satisfy the three 
equations (53) and (54). 

Ifz=f(2,y),¢@=f,(@, y), 6 =f,(@, y) form a system of solutions 
of these three equations, the function f(a, y) also satisfies the 
equation (52), which is a consequence of these three relations. 


* In fact, if the elimination of a and 6 led to another relation © (zx, y, z, p, gq) =0 
different from #'=0, the two simultaneous equations F=0, 6=0 would have a com- 
mon integral V=0 depending upon two arbitrary parameters a and b, which is 
impossible (§ 78). The given integral would therefore depend in reality upon only a 
single parameter. 


V, § 82] EQUATIONS OF THE FIRST ORDER 237 


Conversely, if f,(#, y) is an integral of the equation (52), the 
three equations (53) and (54) are consistent when we replace z 
by f,(@, y), and p and q by the partial derivatives of f(x, y). Hence 
we can derive from them as values for a and 8 two other functions 
a=f,(a, y), b=f,(, y), which form with f(x, y) a system of 
solutions of the equations (53) and (54). 

The new problem, although apparently more complicated than the 
original, is easily solved. In fact, if we differentiate the relation 
(53) with respect to # and to y, regarding now 2, a, 6 as unknown 
functions of « and y, the relations obtained reduce, by (54), to the 
two equations 


OV 0a . OV 0b OVda , OV Ob 
Ga de + Ob ae" Ba dy + Ob By 
and the system formed by the equations (53) and (55) is equivalent 
to the system formed by the equations (53) and (54). 

We see at once that this system is satisfied by taking for the un- 
known functions a and 0 any two constants. This gives as the value 
of z the integral already known, which Lagrange called the complete 
integral. In order to treat the problem in a general way, let us 
observe that the equations (55) are linear and homogeneous in 
0V/ea, 0V/eb. Hence the three equations (53) and (55) are satisfied 
if we set 


(56) — V=0, 


(95) 0, 


Laplaych 
Ele ile Gb 
If these three equations are consistent, they define three functions 
z, a,b of the two variables x and y. This gives an integral z = f(a, y) 
of the equation (52) which does not depend upon any arbitrary 
parameter, and which is commonly called the singular integral. 
If ¢V/éa and 0V/éeb are not zero simultaneously, the equations 


(55) give Cae 
Be a) 6 oy 

which proves that there exists between the functions a and 6 at least 
one relation independent of x and of y. If there exist two relations 
of that kind, a and 6 reduce to constants, which gives again the com- 
plete integral. If there exists only one relation between a and 4, at 
least one of the functions a and 6 does not reduce to a constant. 
Assuming that @ is not constant, we can write the relation between 
a and 6 in the form 


(57) b= $(2) 


0. 


238 PARTIAL DIFFERENTIAL EQUATIONS [V, § 82 


and the two equations (55) become 
Calc Veme Ve. Cu LOVEE Ow ee 
falar ta 8 [=o le + a, s\(@|=0 


Since a is not a constant by hypothesis, these two relations reduce 
to a single relation, and the three equations 


OV OV 
(58) V(x, y, 2, a, 6) = 0, b= (a), a ot op PA 


define a new system of solutions of the equations (53) and (54). In 
particular, the function z = f\(a, y) defined by (58) is an integral of 
the given equation (52). It is evident that this integral depends upon 
the arbitrary function ¢(a). We shall call it the general integral. 
In order to obtain the relation between x, y, z, the arbitrary 
parameter a must be eliminated from the two equations 


OV OV 
(58') algo, Y, %, a, (a) | — 0, Ani 56(@) $'(a) = 0. 


This elimination can be made only after the function ¢(@) has been 
chosen, but the equations (58') always enable us to express two of 
the codrdinates of a point of an integral surface as functions of a 
third codrdinate and of a parameter a. 

The preceding method is related in a very simple way to the 
theory of the surface envelopes. Consider, in fact, the family of sur- 
faces S which represent the complete integral (53) and which depend 
upon two constants a and b. If-we choose an arbitrary relation of 
the form ) = ¢(a) between the two parameters a and 0, we obtain a 
family of surfaces which depend upon only one parameter a, and the 
envelope of this family of surfaces is obtained precisely by eliminat- 
ing a from the two equations (58'). The process by which we deduce 
the general integral from the complete integral consists, therefore, in 
taking the envelope of a one-parameter family of complete integrals 
obtained by choosing an arbitrary relation between the two param- 
eters a and &. Similarly, the singular integral is obtained by taking 
the envelope of all the complete integrals, as the two parameters a 
and 6 vary independently * (I, § 212, 2d ed.; § 220, 1st ed.). 


* We have seen above (§ 71) that all considerations founded on the theory of 
envelopes in the study of differential equations are quite troublesome. All the diffi- 
culties pointed out in the study of the singular solutions of an ordinary differential 
equation of the first order arise again for partial differential equations of the first 
order, The final conclusion is just as before: a partial differential equation of the 
first order, given a priori, does not normally have any singular integrals. This 


V, § 82] EQUATIONS OF THE FIRST ORDER 239 


It would seem from what precedes that we ought to distinguish 
three categories of integrals: the complete integral, the general inte- 
gral, and the singular integral. But Lagrange’s theory itself shows 
that there exist an infinite number of complete integrals. Indeed, 
if we establish between the two parameters a and 0 a relation of a 
definite form b = m(a, a’, b'), containing two constants a! and 0’, the 
corresponding general integral will depend upon these two constants 
a', b', and may be considered as a new complete integral. The 
original complete integral will now be included in the general inte- 
gral, and will correspond to the relation b = w(a, a’, b') established 
between the two parameters a' and 0’. There is, therefore, no essential 
distinction between the general integral and the complete integral. On 
the contrary, the singular integral, as can be seen from its geometric 
meaning, does not depend upon the choice of the complete integral. 

Haeample 1. Consider the generalized Clairaut’s equation 


a= petaqy +f(P, 9) 
It is easily seen that it has a complete integral of the form 

z= ax + by + f(a, 6). 
This complete integral is represented by a family of planes which de- 
pend upon two arbitrary parameters a and 6. These planes envelop 
a non-developable surface 3, which is the singular integral surface of 
the given equation. In order to obtain the general integral, we must 
choose an arbitrary relation between a and 6, say 6 = ¢$(a), and we 
must find the envelope of the planes thus obtained. This envelope, 
which is represented by the two equations 


Waar, natok 
=urtydO)+AGo@O) e+ Ot +e@ 
is a developable surface tangent to the surface & all along a curve I. 
It is evident that we can choose the arbitrary function ¢(@) in such 
a way that the curve of contact [shall be any preassigned curve on 3. 

Example 2. Consider the equation 


7 =f(P), 


of which a complete integral is 


z=ax+f(a)y+ 6. 


p'(a) = 0, 


conclusion does not contradict the reasoning of the text, for we have assumed that 
we can apply the theory of implicit functions to the system of three equations (56), 
and the conclusions are correct only when that condition is satisfied. (See the paper 
by Darboux, Sur les solutions singuliéres des équations aux dérivées partielles du 
premier ordre (Mémoires des Savants étrangers, Vol. XXVII).) 


240 PARTIAL DIFFERENTIAL EQUATIONS [V, § 82 


This equation represents a plane, and the general integral, which is 
given by the system of two equations 


(59) z=antyfa)+e@, V=et+yf'(a)+¢), 
is represented by developable surfaces, which can be defined geomet- 
rically in a very simple way. Draw through a fixed point of space 
(for example, the origin) the planes parallel to the planes which form 
the complete integral; these planes depend only upon the parameter 
a, and consequently envelop a cone (7) whose vertex is at the origin. 
It follows that the edge of regression of the developable surface (59) 
has its osculating plane constantly parallel to a tangent plane of the 
cone (7). Hence the generators of this surface are parallel to the gen- 
erators of the cone just mentioned (I, § 227, 2d ed.; § 224, 1st ed.). 

The equations (56), which determine the singular integral, are in 
this case inconsistent, for the last reduces to 1 = 0. There is: there- 
fore no singular integral. 

Example 3. Consider a family of spheres with a given radius R, 
whose centers remain in a fixed plane. These spheres depend upon 
two arbitrary parameters, and if we take a system of rectangular 
axes with the fixed plane for the xy-plane, they are represented by 


h ti 
the equation CN Cy eter ee 
The corresponding partial differential equation is obtained by elimi- 
nating a and 6 from this equation and the following two, 

x—-a+tpz=)0, y—b+qze=0, 
which gives the equation 

Ad+p'4+7)2-R=0. 

Geometrically this equation expresses the fact that the portion of 
the normal included between any point of the surface and the xy- 
plane is constant and equal to R. The general integral is a tubular 
surface, the envelope of a sphere of radius R whose center describes 
an arbitrary curve in the azy-plane. There is a singular integral 


surface formed by the two planes z =+ R. It is evident that these 
two planes are tangent to all the other integral surfaces. 


83. Lagrange and Charpit’s method. To sum up the preceding, in 
order to determine all the integrals of an equation of the first order, 
(60) F(a, Y; * P; 7) = 0, 
it is sufficient to know a complete integral, that is, an integral depend- 
ing upon two arbitrary constants, In order to determine a complete 


V, § 83] EQUATIONS OF THE FIRST ORDER 241 


integral, let us suppose that, by any means whatever, we have 
obtained another function ®(a, y, 2, p,q) such that the two equations 


(61) F=0, b=a 


can be solved for p and qg, and form a completely integrable system, 
for any value of the constant a. If this is the case, then by solving 
the two preceding equations for p and q, and substituting these 
values of p and q in the equation dz = pdx + qdy, we obtain a com- 
pletely integrable total differential equation 


(62) dz= f(a, y, 2,a)dx + o(a, y, 2, a) dy. 

The integration of this equation introduces a new arbitrary constant 
6, and in this way we obtain an integral of the given equation which 
depends upon the two arbitrary constants a and 0. 

Lagrange and Charpit’s method of integration consists precisely 
in adjoining to the equation F = 0 another equation ® = a such that 
the system (61) formed by these two equations is completely inte- 
grable. For this it is necessary and sufficient (§ 81) that [F, ®] = 0, 
that is, that 


O® O® 
(68) P2408 2+ (Pp +0 e (KX +p2)5-—(¥ +92) 5° =0, 


where, for erate we have set 


a 3 : == Op’ 
The auxiliary function @(a, y, 2, p,q) must therefore satisfy a linear 
partial differential equation in five independent variables. The inte- 
gration of this linear equation reduces in turn to that of the system 
of ordinary differential equations 


(64) EEE ay aU et a Die ari 
de Q PatQq X+pZ VYraZ 


But, for the purpose which we have in view, it is not necessary to 
find the general integral of this system (64) ; it is sufficient to know 
one first integral @ = a of this system, such that we can solve the 
two equations F = 0, =a for p and ¢. 

We can therefore state the following general rule: 


To obtain a complete integral of the equation (60), we first find one 
Jirst integral @ =a of the auxiliary system (64) for which the Jaco- 
hian D(F, ®)/D(p, q) ts not zero; then we solve the two equations 
F=0,@=a for p and q. Substituting the expressions obtained for 
p and q in the equation dz = pdx + qdy, we obtain a completely 


242 PARTIAL DIFFERENTIAL EQUATIONS [V, § 83 


integrable total differential equation. The general integral of this 
equation contains a second arbitrary constant b, and is a complete inte- 
gral of the equation (60). 


We know in advance one integral of the equation (63); that is, the 
function F itself. This integral cannot be used directly, but the 
knowledge of it reduces the integration of the system (64) to the inte- 
gration of a system of three differential equations of the first order. 
The precise nature of the problem to be solved is thus made clear. 

When the function / does not depend upon the unknown function 
z, we may also suppose that the function ® does not depend upon 2, 
and the condition that the system (61) be completely integrable 


is then 
(F, &) =0 
or 
o® o® 0® o® 
! Siac? A ae eee ga ee eS ard ae 
(63°) Her lice a ree 0. 
Hence the auxiliary system (64) takes the form 
dx dy —dp —dq 
! SS Ss ss = 
Ge 'e Q xX Vg 
If we know a first integral 6 = a of this system for which 
D(F, ®) 
D(P; %) 


is not zero, we are led to a total differential equation of the form 


dz = f(x, y, a) da + b (a, Y) a) dy, 
which is integrable by a quadrature. The difficulty of the second 
part of the problem is therefore diminished in this case. This is also 
true of the first part, for we know a first integral F = C of the sys- 
tem (64'); we can therefore replace this system by a system of two 
differential equations of the first order. 


Example 1. Let us consider an equation containing only one of the three 
variables x, y, z (for example, the variable y) : 


Fy, p, 7) =9. 
In this case XY = Z = 0, and the equations (64) give the integrable combination 
dp = 0. Hence the two equations F(y, p, g) = 0, p = a form a completely inte- 
grable system, as is easily verified. For if we solve the given equation for gq, 
the total differential equation to be integrated takes the form 


dz=adx+ f(y, a)dy. 


Hence we obtain a complete integral by a quadrature : 


z=art { fy, a) dy +b. 


V, § 83] EQUATIONS OF THE FIRST ORDER 243 


Example 2. An equation of the form F(z, p,q) = 0 can be reduced to the 
preceding form by taking y and z for the independent variables, but we can dis- 
pense with this change of variables. For in this case we have X = Y = 0, and 


the equations (64) give 
Ne ey 


pegs 
whence a first integral is g = ap. From the two equations 


p qd = ap, F(, p, ¢) = 0 
we then derive 


p=f%,4%), q=af(z, 4), 
and the total differential equation 
dz = f(z, a) (dx + ady) 
can be integrated by a quadrature : 
dz 
=c+ay+b. 
SFea 


Consider, for example, the equation pg —z = 0. Adjoining to it the equation 
gq = ap, we derive from them 


z z z 
p= q=ayf’, de =|" (ar + aay); 


hence a complete integral is given by the equation 


4az = (x + ay + 5)’, 


which represents a family of parabolic cylinders tangent to the zy-plane along 

the entire length of a generator. The zy-plane represents a singular integral. 
The equations (64), in the case where F = pq — 2, have also the first integral 

p—y=a. Starting with this integral, we are led to the total differential 


equation zdy 
’ 


ga d 
2=(y+a) RRL Ser 


which can also be written in the form 


de = a( 3 ). 
y+a 


This furnishes a new complete integral z = (y + a)(x + b), which represents a 
family of hyperbolic paraboloids tangent to the zy-plane. 

Example 3. Let the equation be of the form f(x, p) —f,(y, g) = 9. The dif- 
ferential equations (64) 


Come OU an er ODI Og 
af” Lah OF BF, 
op oq Ox oy 
have the first integral f(z, p) = a. If we adjoin this equation to the given equa- 
tion, we derive from the two relations 


S@ psa, Sy, @):="e, 


the values for p and q, p= ¢(a, a), = $,(y, @),. and the total differential 
equation 
dz = o(a, a)dx + o,(y, a) dy 


244 PARTIAL DIFFERENTIAL EQUATIONS [V, § 83 


can be integrated by two quadratures as follows : 


z= [oe a) dx + foi, a) dy +b. 


When an equation of the first order is of the preceding form, we say that 
the variables are separated. For example, let us consider the equation 


pa — zy = 0, 
which can be written in the form 
DSN 
Zz q 


Equating these two quotients to a constant a, we obtain the total differential 
equation y 
dz = axdz + qo 


whence a complete integral is 


Example 4. Let us propose to find the functions F(z, y, p, q) for which the 
equations (64) have the first integral py — gz = a. For this it is necessary and 
sufficient that the relation pdy + ydp — qdx — xdq = 0 shall be a consequence 
of the relations (64’) ; that is, that the function F shall itself be an integral of 
the linear equation 


dx _ dy dp _ dq 


has the three first integrals 
e+y=C, p+g@=C, py—q=C”, 
and the function F is therefore of the form F (py — qz, x? + y?, p? + q?). The 


investigation of the equation F = 0 for a complete integral is therefore reduced 
to the integration of two simultaneous equations of the form 


P+P=fwW+y,py— qe), py—q=a. 
Making use of the identity 
(p? + q?) (x? + y?) = (py — qu)? + (pe + ay), 
we derive from the two preceding equations 
pet qy =V (0? + y) fe? +, a) — @ = $ (x? + y?, a). 
Solving for p and q, we obtain the values | 


__ ay + xp (x? + y*, a) _ — ar + yp (27 + y?, a) 
i e+ y? pe 2+ y? 


whence we obtain a complete integral by a quadrature, 


yb y p (u, a) 
bape aaretan(¥) 4 (20.0 du + b, 


Pp 


where u = x? + y?. 


V, § 83] EQUATIONS OF THE FIRST ORDER 245 


It is sometimes possible to find a priori, by geometric considerations, certain 
integrable combinations of the differential equations (64). Suppose, for exam- 
ple, that we wish to find the surfaces S whose tangent plane at any point M 
meets at a constant angle V the plane passing through M and Oz. It is clear 
that if a surface S satisfies this condition, all the surfaces obtained from it by 
a helicoidal movement around the z-axis, for which the pitch of the helix is 
equal to h, will also satisfy the condition. Hence the surface envelope = will 
also be an integral of the same equation. This envelope = is evidently a 
helicoidal surface of pitch h. Since we may translate it any distance what- 
ever parallel to the z-axis, it follows that the partial differential equation of 
the problem and the partial differential equation of the helicoidal surfaces 
py — gz = a (§ 72) have, for any value of a, an infinite number of common 
integrals which depend upon an arbitrary constant. Consequently the differ- 
ential equations (64) corresponding to the partial differential equation of the 
surfaces S have a first integral py — gz = a, and the complete integral can be 
obtained by a quadrature. 


Note. It should be noticed that it is not necessary that the relation (63) 
shall be identically satisfied in order that the system (61) be completely inte- 
grable ; it is sufficient that it be satisfied by virtue of the relation F = 0 itself. 
We can sometimes make use of this fact in the search for the function @. In 
fact, the problem of finding an integrable combination of the equations (64) 
reduces essentially to that of finding five functions Az, dy, Az, Ap, Ag Of the 
variables z, y, Z, p, g such that 


Azdxe + A dy + r,dz + Apdp + A,dq 
shall be an exact differential d@ and such that we have also 
Prx + Qhy + (Pp + Qq) x — (X + PZ) Xp — (¥ + 9Z) rq = 0. 


If this last equation is not satisfied except by virtue of the equation F = 0, the 
function ® is not, properly speaking, a first integral of the system (64). How- 
ever, since the multipliers Az, A,,--- are equal to the partial derivatives of #, 
the two equations F = 0, é = a still form a completely integrable system, for 
the equation (63) is then a consequence of F = 0.* A similar remark applies to 
the sytem (64’). 


* When the equation #’'=0 can be solved for one of the variables x, y, z, p, g, we 
may suppose that the function ® does not contain that variable, and it will also not 
appear in any of the coefficients X, Y, Z, P, Q. For definiteness, let us take an equa- 


tion of the form p+H (x, Y, 2, q)=0. 


To find a complete integral, we need only adjoin another equation ¢ (7, y, z, g)=a, 
which forms with the first a completely integrable system. In this case the condition 
[p+/, $]=0 takes the form 


0b Of Ob, (, Of _ ee aie 
On” aq malties t 0z oy” oz] 0g r 


_in which the letter p does not appear. 
More generally, let us suppose that we can satisfy the relation #’=0 by putting 


p=f (x, Y, 2, r), q=$¢ (x, Y, 2, r), 


’ 


246 PARTIAL DIFFERENTIAL EQUATIONS [V, § 84 


84. Cauchy’s problem. Given an equation 


(65) P=f(%Y % @) 
in which the right-hand side is analytic in the neighborhood of a 
system of values (2,, Y5) %» %), and a function ¢(y) analytic in 
the neighborhood ‘of the point y,, such that we have ¢(y,)=%, 
¢'(Y,) =» we proved in § 25 that this equation has an analytic 
integral in the neighborhood of the point (#,,y,) which reduces to 
the given function ¢(y) for «=«a,. Let C be the plane curve rep- 
resented by the two equations « = x,, = (y). Geometrically this 
result may be stated as follows: There exists one and only one ana- 
lytic integral surface of the equation (65) passing through the curve C. 

This proposition is capable of generalization. Let us first consider 
an equation of any form, 


(66) F(x, y, % p, 7) = 9, 
and let us propose to determine an integral surface passing through 
a plane curve, such as C, which lies in a plane « = 2, parallel to the 
yz-plane. Let 2 = $(y) be the equation of the cylinder which pro- 
jects C upon the yz-plane. Since the function ¢ is analytic in the 
neighborhood of the point y,, the equation 


(67) F(x), Yoo “o> P» Yo) = Q, 


where 2, = $(¥Y)) % = $ (Y) and where we regard p as the unknown, 
has a certain number of roots. Let p, be one of them. If the func- 
tion F is analytic in the neighborhood of the system of values (a, 
Yor %p» Por Yo)» and if also the partial derivative (GF /ép), is not zero 
for this system of values, the equation (66) has a root p = f(a, y, 2, q) 
which is analytic in the neighborhood of the system of values (a,, y, 
%» Yo) (1, § 193, 2d ed.; § 187, Ist ed.). Hence we are led back to 
an equation of the form (65), which shows that the equation (66) 
possesses an integral surface through C. As a matter of fact, the 
reasoning proves that this equation has m integral surfaces which 
satisfy the conditions if the equation (67) is of degree m with 
respect to p. There is no possible exception unless one of the roots 


where \ denotes an auxiliary parameter. We need only replace X by a function of 
x, y, z such that the equation dz=fdx+¢dy is completely integrable, which again 
leads to a linear equation for A (x, y, Z): 

(2.27) 


Of, Of 44 Of (OX, Or ,\_G6  0¢,, 06 
hs alas 7 t= ost pelt oe ox oz 


(ANTOMARI, Bulletin de la Société Mathématique, Vol. XIX, p. 154.) 


V, § 84] EQUATIONS OF THE FIRST ORDER 247 


of the equation (67) satisfies also the relation ¢F'/ép = 0 at all the 
points of C, since x, y,, %, are the codrdinates of any point of 
this curve. 

Let us consider finally any curve I, represented by a system of 
two equations 


(68) e=(y), 2=H(Y); 
and let it be required to determine an integral surface of the equa- 
tion (66) which passes through IT. This problem, in turn, can be 
reduced to the preceding by means of a change of variables; for if 


Beet eer Yaa ety ty 
the relation dz = pdx + qdy becomes 

dz = pdX + pr'(Y)dY + qayY, 
and from this we derive 


pase PMY) +9 = 5% 
The equation (66) is then replaced by the equation 
@) r[x+ac, nad 2—x—yZ]=0 
and it remains to find an integral of this new equation which 
reduces to w(Y) for X = 0. Hence we see that in general an inte- 
gral surface of an equation of the first order is determined if we 
assign a curve lying on that surface. There may be several integral 
surfaces satisfying this condition if the equation similar to (67) has 
several distinct roots, just as an ordinary differential equation of the 
first order and of degree m in y’ has in general m integral curves 
passing through a given point. We shall return later to the excep- 
tional case in which this reasoning fails. 
The problem of determining an integral surface of a partial differ- 
ential equation of the first order through a given curve has been 
called Cauchy’s problem. This name is used to remind us of the 
close relation just explained existing between this problem and 
Cauchy’s general theory. We shall now show how Cauchy’s problem 
can be solved by an elimination if we know a complete integral, and 
this will furnish also a verification of the preceding results. 


Let V(x, Y, 2, a, 6) = 0 


be a complete integral, and let T be a given curve not situated upon 
the singular integral surface nor upon one of the integral surfaces 


248 PARTIAL DIFFERENTIAL EQUATIONS [V, § 84 


obtained by giving to a and to 6 constant values. Cauchy’s problem 
reduces to determining the function ¢ (a) in such a way that the given 
curve T shall lie upon the surface S defined i the two equations 


aan 
(69) V (a, Y, *) a, (a) | == 0), +54 B d' (a) = = 


Let us suppose that the coordinates x, y, 2 of a point of I are 
expressed as functions of an auxiliary parameter A, 


(70) a=f,A), y =f,A); z=f,(), 
and let U(A, a, 6) be the result obtained by replacing a, y, 2 in 
V (x, y, 2, a, b) by the preceding expressions. The two simultaneous 
equations 


(71) U[A, a, 6(a)]=9, oe 


a+ gap 0) = 0 


determine the values of A and a which correspond to the points of 
intersection of the curve I with the surface S. If the surface S 
passes through the curve TI, these two equations form an indeter- 
minate system. Hence, eliminating from these two equations, we 
obtain an identity. This elimination leads to a relation between a, 
$(a), $a), 
(72) Il La, $(@), (4)] = 9, 

that is, to a differential equation of the first order for the determina- 
tion of (a). It would seem, therefore, that the problem has an 
infinite number of solutions, contrary to Cauchy’s result. But it is 
easy to deduce from the equations (71) another relation not contain- 
ing ¢'(a). In fact, let us suppose that the curve I lies entirely on 
the surface S. When a point moves on I, a is a function of X which 
satisfies the two equations (71) simultaneously. Hence, if we differ- 
entiate the first of these two equations with respect to X, it follows 
from this result and the second that ) 


0U 
(73) Nae 0. 
This equation contains only A, a, ¢(a). Eliminating » from the two 
equations U = 0, dU /éX = 0, we obtain an equation which determines 
the function ¢(a). The method to which we are led has an evident 
geometric meaning. In fact, the equation U(A, a, 6) = 0 determines 
the values of X which correspond to the points of intersection of the 
curve [ with the complete integral. If we also have 0U/0A = 0, this 
equation has a double root, and the complete integral is tangent to I. 


V, § 85] EQUATIONS OF THE FIRST ORDER 249 


Eliminating A from the two equations U(A, a, b)= 0, 0U/exA = 0, 
the condition obtained, (a, 6) = 0, therefore expresses the fact that 
the complete integral is tangent to I, and the desired integral surface 
through T may be defined as the envelope of the complete integral 
surfaces tangent to the curve T. This result is geometrically almost 
intuitive.* 


85. Characteristic curves. Cauchy’s method. Cauchy’s method is 
independent of the theory of the complete integral. We shall now 
present it in a geometric form. For this purpose, let us first consider 
the meaning of a non-linear partial differential equation 


(74) F(x, ¥y, %, Pp, = 9. 

This equation may be regarded as a relation between the direc- 
tion cosines of the tangent plane to an integral surface S through a 
given point (a, y, #) of space. Hence this tangent plane cannot be 
any plane passing through the point (a, y, z). Since the possible 
tangent planes form only a one-parameter family, they envelop in 
general a cone (7) whose vertex is the point (a, y, z). It follows 
that the tangent plane at any point M of space to each integral surface 
S passing through this point is also tangent to a certain cone (T) 
whose vertex is at M. 

The cone (7) depends, of course, upon the function F, and also 
upon the position of its vertex. In order to obtain the equation of 
the cone (7) whose vertex is (a, y, ), we must, by its definition, 
find the envelope of the planes 


(75) Z—z=p(X—«2)+4q(V¥—y), 


where the parameters p and g are connected by the relation (74). We 
must therefore eliminate p and g from these two equations and the 
new relation (I, note, § 208, 2d ed.; § 202, Ist ed.) 


(76) (VFX Fao. 


* It is easy to obtain the general integral of the differential equation (72). In fact, 
if we replace \ by an arbitrary constant Xo, the function ¢ (a) defined by the equation 


(e) U [Ao, a, p (a)] =0, 
also satisfies the equation 


; oU 
(c’) oU 


—— + 
Ca 0¢(a) 
Hence ¢ (a) satisfies also the equation obtained by eliminating Xo from (e) and (e’), 
but the resulting equation is exactly the equation (72). The relation (e) therefore 
represents the general integral of the equation (72). There is also a singular inte- 
gral, which is indeed precisely the desired solution of the given problem. 


p’ (a) =0. 


250 PARTIAL DIFFERENTIAL EQUATIONS [V, § 85 


The two equations (75) and (76) represent the characteristic direc- 
tion, that is, the generator of the cone (7) which is the line of con- 
tact of the tangent plane. If we suppose that the axes of codrdinates 
are rectangular, we can obtain immediately the equation of the 
normal cone (NV), which is generated by the normals to the different 
integral surfaces passing through the point M. For, since the equa- 
tions of the normals are 


X—a+p(Z—z2)=0, Y-yt+q(Z—-2)=9, 


the elimination of p and q gives the equation of the cone (1) in 
the form : 


X—2 Y— 
(77) F(», Y; a eee 0. 


If the given equation (74) is linear in p and q, the cone (JV) is a 
plane and the cone (7) reduces to a straight line A. We have seen 
(§ 76) that the integration reduces in this case to the search for the 
curves which are tangent in each of their points to the correspond- 
ing straight line A. We are led to Cauchy’s method by extending 
this process to non-linear equations. 

Let S be an integral surface represented by the equation 


z 17 (O08), 

At each point M of this surface the tangent plane is also tangent 
to the cone (7) along a generator (G). We shall give the name char- 
acteristic curve to every curve C of the surface S which is tangent 
in each of its points to the corresponding generator G. Through 
each point of S (excepting the singular points, if there are any) 
there passes one and only one curve of this kind. The name charac- 
teristic curves will be justified later (§ 86). 

The key to Cauchy’s method is that we can determine these curves 
by a system of ordinary differential equations without knowing the 
function f(x, y). In the first place, the tangent to the curve C coin- 
cides with the straight line G represented by the two equations (75) 
and (76), which may be written in the form 


in the notation of § 83. Along a characteristic curve 2, y, 2, p, 4 
are functions of a single independent variable, and we may write the 
relations between the differentials dx, dy, dz in the form 


He PQ" P+ay 


V, § 85] EQUATIONS OF THE FIRST ORDER 251 


where w is a conventional auxiliary variable which is introduced 
merely for symmetry. Along this curve C we have also 


dp = rd« + sdy, dq = sdx + tdy, 


where 7, s, ¢ are the usual second derivatives of the function f(a, y). 
On the other hand, since z= f(a, y) is an integral of the given 
equation (74), the partial derivatives r, s, ¢ also satisfy the two 
relations 


X+pZ+Pr+Qs=0, Y+qZ+Ps+ Q=0, 


which are obtained by differentiating (74) with respect to # and 
with respect to y. Replacing the differentials dx and dy by Pdu and 
Qdu respectively, the expressions for dp and dq become 


dp = (Pr + Qs) du, dq = (Ps + Qt) du, 
or, using the preceding relations, 
dp =—(X + pZ)du, dg =—(Y + qZ)du. 


Adjoining these equations to the equations (78), we arrive at a 
system of ordinary differential equations 


Pee a ek Ost Mike ae OR A beg. t 
Je Q Pet+Aaq X+pZ VYrqZ 


which is identical with the system (64) to which we are led by 
Lagrange’s method. 

This system of differential equations is absolutely independent of 
the integral considered. We derive from it the following conclusions : 
Let («,, y,, %) be the codrdinates of a point I, of S, and let p, and 
q, be the values of p and g for the tangent plane at this point. If 
the function F is analytic in the neighborhood of this system of 
values, and if not all the denominators of the quotients (79) vanish 
simultaneously for x,, Y) %» Por Yor the equations (79) have one and 
only one system of integrals which take on the values ~,, y,, 25 P53 > 
for w=0. It follows that if two integral surfaces are tangent at a 


(79) 


du, 


point (8) Yor %)) they are tangent along the entire length of a common 
characteristic curve through that point. 

For convenience we shall call every system of values assigned to 
the five variables x, y, z, p, g an element. Thus, an element may be 
thought of as consisting of the set of a point whose codrdinates are 
(x, y, #) and a plane through that point whose position is defined 
by the values of p,g. Along an entire characteristic curve, x, y, 2, p, 
and g are functions of an independent variable vu. To each point of 


252 PARTIAL DIFFERENTIAL EQUATIONS [V, § 85 


a characteristic curve there corresponds, therefore, an element com- 
posed of this point together with the plane through this point defined 
by the values of py and g. But from the equations (79) we have 


dz dx d 

ig ae 

so that this plane contains the tangent to the curve at the point 
(x, y, #). When the point (a, y, #) describes the characteristic curve, 
the corresponding plane envelops a developable surface passing 
through this curve, which is called the characteristic developable 
surface. Thus, to each characteristic curve there corresponds a char- 
acteristic developable surface through that curve. We shall hereafter 
use the words characteristic strip to denote the combination of the 
curve and the developable surface, and we shall refer to the curve 
as the characteristic curve, to avoid any possibility of ambiguity. 
With this understanding, a characteristic strip is composed of an 
infinite number of elements which depend upon an independent 
variable, and the infinitesimal variations of a, y, z, p, g are con- 
nected by the relations (79). A characteristic strip is therefore 
completely defined if we are given one of its elements, and the 
theorem stated a moment ago can again be expressed in the follow- 
ing exactly equivalent form : 


If two integral surfaces have a common element, they have in com- 
mon all the elements of the characteristic strip to which the given 
common element belongs. 


The totality of all characteristic strips depends upon three arbi- 
trary parameters. In fact, a characteristic strip is determined if one 
of its elements (@,, ¥) 2%» Po %) 18 given. One of the codrdinates, 
x, for example, may be assigned a given numerical value, and, more- 
over, by definition the relation F(a,, y,, %) Po» 7) = 9 is satisfied. 
Hence only three parameters remain arbitrary. 

In order to determine the characteristic strips, let us observe first 
that F = const. is a first integral of the equations (79). Hence, if 
F(a, y, 2, p, 7) vanishes for the initial element (a, y,, 2%» Po» %)» F 
vanishes throughout the entire length of the characteristic strip 
through that element, as we see also from the derivation of the 
equations (79). In order to find the characteristic strips of the given 
equation, we can therefore adjoin to the system (79) the relation 
F = 0 itself, which reduces that system to one of three differential 
equations of the first order. 


V, § 85] EQUATIONS OF THE FIRST ORDER 253 


Let us suppose that we have obtained the equations of the charaec- 
teristic strip in finite terms; and, for definiteness, let 


(80) ie =F (® Xs Yor %» Por Lo)» % = (Ly Ly Yor %» Por Yo) 
P = f, (a, Xo» Yor *o» Po» Yo)» q = f(a, Loo Yor 9» Poo I) 


be the equations of the characteristic strip through the element 


(yy Yor %» Por Uo): 


The two first equations of (80) represent the characteristic curve 
itself, and every integral surface, being a locus of the characteristic 
curves, will be obtained by supposing that «,, y,, 2, 2,» 7, are func- 
tions of an auxiliary parameter v. We are therefore led to investi- 
gate how these five functions of v may be chosen in order that the 
surface generated by these characteristic curves shall be an integral 
surface. We shall introduce with Darboux an auxiliary variable w, 
and write the equations in a symmetric form. Let 


(81) f= p,(U, Loy Yor ~o» Poo Yo) 
ame (Us, ys Yo @o) Po on) 


(82) Nailed ean DNs 
1 Oegs $,(u, Xoo Yor ~o9 Po» Qo) 


| = $,(U; Xp Yor %» Por Lo)s 


be the equations which represent the integral of the system (79) 
which takes on the values x,, y, 2%, Po 7, respectively for u = 0. 
If we replace x,, ¥,, 2) Py» YJ) n these expressions by functions of a 
second auxiliary variable v, the equations (81) represent in general 
a surface S, uw and v being regarded as two independent variables. 
In order that the surface S be an integral surface, and that the 
curves v = const. be the characteristic curves, the equations (82) 
must give precisely the values of » and qg which determine the 
tangent plane to that surface; and, moreover, the relation F=0 
must be satisfied at every point of S. Hence the five functions 
x, y, 2, p,q of the two variables wu and v must satisfy the three 
conditions 


(83) F(a, Y> *) Py q) a 0, 
02 Ox Osan 
(84) Eye irs mt Be wae 


(85) — q5-=0. 


254 PARTIAL DIFFERENTIAL EQUATIONS LV, § 85 


Since the five functions ¢; are integrals of the system (79), we have, 
as remarked above, F(a, y, 2, Pp, 7) = F (2) Yoo %» Po» Yo): Hence the 
relation (83) will be satisfied if 


(86) F (Xo) Yor % 2 Por Io) = 9. 
The relation (84) is identically satisfied, for it is a consequence of 
the differential equations (79). Cauchy transforms the condition (85) 
as follows: Indicating by H the left-hand side of (85), and differen- 
tiating with respect to wu, we find 
OF Ge. Ox Py  Ox0p dyog 
du dudv. ? dudv 1 dudv Ov du dv Ou 


On the other hand, differentiating the relation (84) with respect to v, 


we have also 
Oz Cx Cy op Ox a OY. 


C 
reer seand ened WY (amen ue Wn Tf lye 
Cudv P Oudv Z Cudv Ov Cu —s- Ov Ou’ 


whence, subtracting, 


Ou Ov Ou = ov ae Ov du ov Ou 


or, replacing the derivatives with respect to w by their values 
obtained from the relations (79), 

Che oy Op 4 (p 0x oy 

Ou ees a, Specs pie ele 
Finally, observing that the five functions a, y, 2, p, q of v satisfy 
the relation 

F(x, Y> *) P» qV= 0 

and that we therefore have also __ 


Ox oy Op 
xS4yey ee em 


we may write the preceding value of 6H/éu in the form 


Oe Ox Cg van 
ou (» EN SET hae je) =~ 2H 


We derive from this relation the following value for H, 


a 
+0 oS 
(87) 


(88) Hades. 7" 
where H, denotes the value of H for wu = 0, that is, when a, y, 2, p, ¢ 
reduce respectively to %,, ¥, %, Po: Yo Since the function F, and 
consequently also the partial derivative Z, is supposed analytic 
in the neighborhood of the system of values Lor Yor %y» Por Ao» the 


V, § 85] EQUATIONS OF THE FIRST ORDER 255 


necessary and sufficient condition that H be zero is that H, be zero, 
that is, that (¢z/ov), = p,(ex/ev), + 9,(2y/0v),. 


Summing up, to obtain an integral surface * it is sufficient to replace 
Lor Yor Xo» Por Yo wn the equations (80) or (81) by functions of an 
auxiliary variable v which satisfy the two conditions 


Cz Ox 01 
(89) F(x,, Yoo ~o9 Poo %) = 0, a =D, ae +q,—?: 


This method leads very easily to the solution of Cauchy’s problem. 
In fact, if we wish to determine an integral surface through a given 
curve I’, we may take for @,, y,, z, the codrdinates of any point of 
that curve expressed as functions of a variable parameter v, and the 
equations (89) then determine p, and g,. The solution may also be 
stated in geometric language as follows: The first of the equations 
(89) expresses the fact that the plane through the point (~,, y,, z,) 


* The argument presumes, however, that the denominators P, Q, X+pZ, Y+qZ 
are not all zero for the initial values 2%, Yo, 2, Po, Yo. In case they are, the equa- 
tions (81) and (82) reduce to x=, y= Yo, Z2= 2%, P= Po» Y= Yo, Whereas if we suppress 
the auxiliary variable uw, the equations (79) may have integrals which take on the 
given initial values (§ 31, Note). Hence the integrals of the given equation which 
satisfy also the four equations 


P=0, Q=0, X+pZ=0, Y+qZ=0 


are not given by the general method. Such integrals, if there are any, are singular 
integrals. There exist normally no such integrals for an equation given a priori and 
not formed by eliminating constants. 

The reasoning can be arranged so as to put in evidence the hypotheses necessary 
for the validity of the conclusions. Let us suppose first of all that the function 
F(x, y, 2, p, Y) is an analytic function of x, y, z,p, g. In order to show that every 
integral z=/ (x, y) represents a locus of characteristic curves, it is not necessary to 
suppose that that integral is analytic; it is sufficient to assume that it has continuous 
partial derivatives of the second order 7, s, ¢, since only these derivatives appear in 
the proof. The characteristic curves, being defined by a system of analytic differen- 
tial equations, are necessarily analytic curves, and, consequently, on every integral 
surface, whether it is analytic or not, there exists a family of analytic curves, namely, 
the characteristic curves. The functions $1, ¢2, ---, 5, which represent the general 
integral of the equations (79), are analytic functions of wu and of the initial values 
Xo, Yo, Zo, Por Yo (§ 26). In order that the calculations which follow, and their con- 
clusion, be rigorous, it is sufficient that these initial values be continuous functions 
of a parameter v, and that they have continuous derivatives, but it is not necessary 
that they shall-be analytic functions of v. 

This is quite in accord with the method of the variation of constants. If the com- 
plete integral V(x, y, z, a, b) is an analytic function of its arguments, the same 
will be true of F(x, y, z, p, g), but nothing in the argument requires that the arbi- 
trary function b= ¢ (a) shall be an analytic function of a. A similar remark applies 
to the general integral of a linear equation. For more details on this subject see 
E. R. Heprick, Ueber den analytischen Character der Lésungen von Differential- 
gleichungen (Inaugural-Dissertation, Gottingen, 1901). 


256 PARTIAL DIFFERENTIAL EQUATIONS [V, § 85 


determined by the values p, and q, is tangent to the cone (T) whose 
vertex is that point; and the second of the equations (89) expresses 
the fact that this plane passes through the tangent to the curve I. 
Hence the whole process may be formulated as follows: Through 
the tangent at the point M to the curve T pass a plane tangent to the 
cone (T) whose vertex is M ; let C be the characteristic curve through 
the element thus determined ; the surface generated by this charac- 
teristic curve, as the point M describes the curve T, is an integral 
surface through the curve T. 

There will be as many surfaces fulfilling these conditions as there 
are tangent planes to the cone (7) through a tangent to the curve I. 
It is also clear that we should associate tangent planes which form 
a continuous sequence. 

Let us consider first the general case where the tangent to the 
curve Tis not a generator of the cone (7). Since p, and gq, fix the 
position of the tangent plane to the cone (7), the direction cosines 
of the element of contact of (7) with that plane are proportional to 
Py Qo Po Po + 27 by the formule (75) and (76). Since the differ- 
ence P,(¢y,/dv) — Q,(ex,/ev) is not zero, the values of p, and of q, 
derived from the equations (89) are analytic functions of v in the 
neighborhood of the given point of I. On the other hand, we can 
solve the first two equations of (81) for wu and v, for the functional 
determinant 0x/0u dy/dv — dy/du dx/ov reduces for u = 0 to 


(aa) ar Me) 
Ou] \ dv Ou] \ dv ‘ 


that is, to P,(@y,/ev) — Q,(éx,/ev). Substituting these values of w and 
v in the third of the equations (81), we see that 2 is an analytic 
function of # and y in the neighborhood of the given point (see § 84). 


If the tangent at a particular point of the curve I coincides with the element 
of contact of (7) with the plane determined by the values po, g, at that point, 
this point is in general a singular point for the corresponding integral. If the 
same thing happens at every point of T, we must distinguish two cases, according 
as the curve I is a characteristic curve or not. 

If the curve I is a characteristic curve, it is tangent at each of its points to an 
element G of the cone (T) whose vertex is at that point, and the characteristic 
developable surface is the envelope of the tangent plane to the cone (7) along 
the generator G when the vertex M describes the curve T. The characteristic 
curve through each of the elements thus determined coincides with the curve 
Yr itself, and the equations (81) do not define a surface. But it is clear that in 
this case the problem is indeterminate. For let M be a point of I, let P be the 
plane tangent to the cone (T) whose vertex is M along the tangent G tol, and 
let I’ be another curve through M whose tangent at M is a straight line of the 


V, § 85] EQUATIONS OF THE FIRST ORDER 257 


plane P different from G. From what we have just proved, the integral surface 
through I’ contains the curve I. 

If the given equation (74) is not linear in p and q, as we shall suppose, the 
curve I‘ can be tangent at each of its points to a generator G of the correspond- 
ing cone (7) without being a characteristic curve. The family of curves 
having this property depends, in fact, upon an arbitrary function. Let 


Y-y 3) 
ee =0 
xX~—-% A—f 


(2, Y, 2; 


be the equation of the cone (T) whose vertex is (x, y, z). In order that a curve 
I’ be tangent at each of its points to an element of (7), the codrdinates z, y, z 
of a point of that curve must be functions of a variable v satisfying the condition 


(90) #(x, 23 oe “,)=0 


If we take z, for example, as the independent variable, we may choose arbi- 
trarily y = f(x), and then, substituting f(z) for y in the preceding relation, we 
have a differential equation of the first order for the determination of z as a 
function of x. Every curve not a characteristic satisfying the condition (90) 
will be called an integral curve. 

Now let us suppose that the curve I, for which we wish to solve Cauchy’s 
problem, is an integral curve. From each point M of I there issues a character- 
istic curve tangent to I, and it follows from the preceding arguments that the 
surface S generated by these characteristic curves is an integral surface. Indeed, 
it is sufficient to take for zp), ¥p, 2) the codrdinates of a point of I’, and for p), 
Yo the coefficients p and q of the plane tangent to (7) along the tangent toT. 
But this curve T is a singular line on the surface S; for if it were not, the 
derivatives r, s, t would have finite values in a point of I’, and, since we have 
Q, dt) = Pydyy, the arguments of page 251 to establish the equations (79) would 
apply without modification, and we should conclude that the curve [ is a 
characteristic curve, which is contrary to the hypothesis. This curve I, which 
is the envelope of the characteristic curves of the surface S, is the analogue of 
the edge of regression of a developable surface. 


Note. Cauchy’s method also leads readily to a complete integral; 
for we can satisfy the conditions (89) by putting a, =a, y,=42, 
z, =, where a, 6, c are any three constants and where p, and q, 
satisfy the relation F(a, }, ¢, p,, q,)= 9. The integral surface thus 
obtained is the locus of the characteristic curves starting from the 
point (a, b,c), which is evidently a conical point for that surface. 
If we regard one of the coérdinates a, b, ¢ as a numerical constant, 
we have a complete integral. 


Example 1. Let us consider the equation treated by Cauchy, pq — zy = 0. 
Making use of the equation itself, we see that the differential equations of the 
characteristic curves can be written in the form 


pdt=qdy= . = zdp = ydq. 


258 PARTIAL DIFFERENTIAL EQUATIONS [V, § 85 


We derive from them successively the integrable combinations 


Cpe Ca ols dz =P 2adx = 12 ydy, 
p x q y x y 
and the characteristic strip through the element (2X), Yo, 29) Pos Yo) 1S represented 
by the equations 
x y 
Bae Fad, am = Pet —af) = 2 vi), 
Po Xo do Yo Xo Yo 
where £9, Yo, Pos Yo are connected by the relation po%) = XpYp. In order to 
obtain the integral which, for « = z,, reduces to ¢(y), we shall put, as in the 
general method, y, = v, 2) = ¢(v). In this case the equations (89) give 


Hi XD) 
= ¢’(v), Dr ° ° 
Jo (v) 0 (0) 


The required integral is therefore represented by the simultaneous system of 
two equations 


Vv 
¢(v) 


which define v and z as functions of x and y. These two equations may be 
replaced by the equations 


z—o(v)= (22 — a) = ©) a v9, 


[z— o(v)]? = (2? — 22) (y2— 02), [2 — o(v)] 9’(v) = 0 (2? — 23), 


of which the second may be obtained from the first by differentiating with 

respect to the parameter v. The desired integral can be obtained by eliminating 

v, and it follows that this result is quite in accord with Lagrange’s theory. 
Example 2. Let us consider again the equation of page 240, 


(1+ p? + g*)2?— R? = 0, 


which states that the length of the segment of the normal cut off by the 
zy-plane is equal to R. Hence, in order to obtain the normal cone (NV) at the 
point M of space, we need only describe about the point M as center a sphere 
of radius R, and then take the cone of revolution whose vertex is M through 
the circle in which the zy-plane cuts this sphere. The corresponding tangent 
cone (T) is the cone of revolution whose vertex is M. We know here a com- 
plete integral, the spheres of radius R having their centers in the xy-plane. 
The characteristic curves, which are the limiting positions of the intersections 
of two spheres that are an infinitesimal distance apart (see § 86), are there- 
fore circles of radius R, whose planes are parallel to the z-axis and whose 
centers are in the zy-plane. Every integral curve, as we have seen, may be 
regarded as the envelope of the characteristic curves on an integral surface. 
These curves are therefore represented by the system of three equations, 


(x — a)? + [y— o(a)]? + 22 — B= 0, 
x— a+ [y— o(a)] ¢(a) =0, 
1+ $2(a) + ¢(a)¢”(a)— y9"(a) =O, 


where ¢ (a) is an arbitrary function. 


V, § 86] EQUATIONS OF THE FIRST ORDER 259 


86, The characteristic curves derived from a complete integral. The concept of 
characteristic curves can be derived in a very natural manner from Lagrange’s 
theory. We have seen, in fact, that if V = 0 is a complete integral of a given 
equation of the first order, we obtain an integral surface by eliminating a from 


the two equations 


oV oV 
91 V [a@, 9 & Uy == 0, rac AL 
(91) [2 V5 % @, 6 (0)] =a asta 


where ¢(a) is an arbitrary function. If we give to the parameter a a constant 
value, these two equations represent a curve whose locus is the integral surface. 
The equations of this curve are of the form 


¢'(a) = 0, 


(92) VAG veyeth, Ola U, —— te ¢ = 0, 
a 


where a, b, c are arbitrary parameters. These curves form a complex, and we 
see that the integral surfaces are generated by the curves of this complex 
associated according to a suitable law. The name characteristic curves is self- 
explanatory, since they are the curves of contact of the complete integral with 
its envelope. 

The characteristic developable surfaces also appear in a natural manner. 
Let us consider a characteristic curve corresponding to the values a), bo, Cy of 
the parameters a, 6, c. All the integral surfaces obtained by means of func- 
tions ¢, such that we have b) = $(d), ©) = ¢’(@), pass through this curve and 
are tangent to each other along this entire curve, for the values of p and g, which 
for any point of an integral surface are given by the relations 


eV oV OV oV 
+p——= +9—= 


93 dic pAlied oT NUM AMS My ESL 
ad ee ; oy 1 


0, 

are the same for all these surfaces. It is therefore natural to associate with 
each characteristic curve a characteristic developable surface passing through 
this curve. The four equations (92) and (93) enable us to express four of the 
variables x, y, 2, p, g in terms of one of them and of the three arbitrary con- 
stants a, b, ec. In order to prove the identity of the forms thus defined with 
those of the characteristic strips deduced from Cauchy’s method, let us suppose 
that the complete integral is represented by an equation of the form 


Z= D(z, y, a, 6). 
The equations (92) and (93) then become 


ob O® 

94 Zi Pia. b), —+—c=0, 
(94) (4d). + 
® om 
(95) a a 
Ox oy 


The relations (94) and (95) enable us to express the five variables (x, y, z, p, q) 
in terms of one of them (zx, for example) and of the three arbitrary constants 
a, b,c. The proof reduces to showing that these functions satisfy the differen- 
tial equations (64). Since the function @(x, y, a, b) is a complete integral of 
the equation F = 0, we have already between these functions the two relations 


(96) F(x, Y; 2, p,q) = 9, dz = pdx + qdy. 


260 PARTIAL DIFFERENTIAL EQUATIONS [V, § 86 


On the other hand, we deduce from the second equation of (94) 


om O27 oh O27 
97 ee ey da +¢ ) dy = 0. 
(@7) (= ox - ob = 2 (5 oy ob oy a 


Now if we differentiate with respect to the constants a and b the identity 


om of 
F(x, Y, ®, on =)= 0, 
we find 
2 2 

Oa phy ae ey 

oa 0a OX 0a oy 

o® O27 O27 
Ti mie HP 0 
Obi ay ae ¢ 


and consequently, by eliminating Z, we have 


o2 o2b oz O27 
98 'P. C——— Cc ==) 
(98) (5 ox ob =| AY (= oy a ob ) 


? 


A comparison of the two relations (97) and (98) shows that we have dx/P=dy/Q. 
The remaining equations of (64) are established as in § 85, by comparing the 


relations 
07h o7@ om oe 
dp = —- dz + ——d dq = —_ dx + —-d 
Fe et ae ay 1 oeay ay?” 
which are deduced from the equations (95), with the relations 
OP o*@ om 
X+Z—+ P — = 
iJ Ox a ox? +@ Ox OY , 
2 g 
bgt peed tay seeoter a4 fe 5) 
oy ox Oy oy? 


which in turn are obtained by differentiating the identity 


with respect to the variables z and y. 


Note. The theory of the complete integral applies to linear equations as well 
as to the non-linear equations. It seems at first sight, on the contrary, that 
Cauchy’s method is altogether different for linear equations and for non-linear 
equations. In fact, the characteristic curves of a linear equation, or of an 
equation which separates into several linear equations, form a congruence and 
not a complex. But if we associate with each characteristic curve a charac- 
teristic developable surface, the paradox disappears. Each characteristic curve 
belongs, in fact, to an infinite number of characteristic developable surfaces 
which depend upon an arbitrary constant, so that this family of characteristic 
strips does depend upon three arbitrary constants. Let us consider, for example, 
the equation of the cones pa + qy—z=0. The equation z = az + by represents 
a complete integral formed by all planes P through the origin. The character- 
istic curves are the straight lines passing through the origin, and the character- 
istic developable surfaces are the planes P themselves. We shall therefore 
obtain a characteristic strip by associating with a straight line through the 
origin a plane through that straight line; this set actually depends upon three 
arbitrary constants. 


V, § 87] EQUATIONS OF THE FIRST ORDER 261 


87. Extension of Cauchy’s method. Cauchy’s method can be extended 
without difficulty to an equation in any number of independent 
variables, 

(99) F(a, Boyt tty Uns % Py) Por? *y Pn) = 0. (r= =) 
Let z = ®(x,, x, -+-, x,) be any integral of the equation (99); we 
shall designate as an element of this integral the set which consists 
of a system of particular values v8, 28, ---, x2 of the independent 
variables, together with the corresponding values 2°, p},---, p° of 
the function ® and its partial derivatives. Let us suppose that an 
element of the integral, starting with certain initial values 2°, 2°, 9, 
varies so as always to satisfy the differential equations 


bo Sashes dx 


(100) a =, ip SS Oc == >! 
; 1 2 n 
where, as in § 83, 
. tod Fe OF 0 
gine. a ar roe Noh 


It is clear that these equations determine completely a family of 
curves (or one-dimensional manifolds) on each integral. For if z is 
known as a function of «,, x,,---, %,, the same thing is true of the 
partial derivatives p;, and consequently of the functions P;. These 
relations (100) form, therefore, a system of (n — 1) differential equa- 
tions of the first order between the n variables (#,, z,,---, x,). By 
the theory of differential equations, through each point of the inte- 
gral surface there passes in general one and only one of these mani- 
folds. If to each point (a,, #,---,x,, #) of one of these manifolds 
we associate the corresponding values of p,, p,,+++; Pa, We have a 
simply infinite sequence of elements, which we may again call a char- 
acteristic strip. We shall show that, without knowing the expression 
for the function z, we can adjoin to the relations (100) other differ- 
ential equations enabling us to define completely the variation of 
the variables x;, z, p, along a characteristic curve. 

Let us start from an element of the integral (a?, 2°, p?), and let us 
consider the characteristic strip through this element. Along this 
characteristic strip the variables x;, 2, p, are functions of a single 
independent variable satisfying the relation / = 0, whose differen- 
tials satisfy the equations (100) and also the relations 


dz = pdx, A= See + Prd dp; = pj,da, ae + DinILy, 


Oz , 
Bo Fe bey, (= 1, 2,-+., 0) 


262 PARTIAL DIFFERENTIAL EQUATIONS [V, § 87 


which result from the definition. Differentiating the relation F = 0 
with respect to the variable «;,, we find 


X,;+D:iZ+ Pipi te: > +P pin = 0. 


Indicating by du the common value of the quotients (100), and 
replacing P; in the preceding relation by dx, /du, we find 


(X;+ p,Z)du + pi, da, + +--+ pydz, = 9, 


that 1s, (X, + p;Z)du + dp; = 0. 


This shows us that the elements of an integral satisfy, along the 
entire length of a characteristic strip, the system of differential 
equations, 


ax, dz — dp, 
(101) a 


= 2 = — ** = du. (i,k =1,2,--- 0 
yas te Pa ete a an ete nate Oe ( te ”) 


These equations do not depend upon the function @; hence we 
can determine the successive elements of a characteristic strip, pro- 
vided that we know a single element (x?, 2°, pz). We conclude from 
this, just as before (§ 85), that ¢f two integrals have a common ele- 
ment, they have in common all the elements of the characteristic 
strip through that element. | 

If, as we shall assume, the denominators of the equations (101) 
remain finite and are not all zero for the initial values, we derive 
from these equations 


(102) Pes d;, (u, xe» a Ph) 


Y= Y(u, ioe oe Dis 


x; =f; (u, Lap 2, Pr) 


where 2°, 2°, »? denote the initial values corresponding to the initial 
value w = 0 of the auxiliary variable w, and where the functions f,, 
¢,, wy are continuous differentiable functions of w and.of the initial 
values, at least within certain limits. 

Since each integral is a locus of characteristic curves, it is clear 
that every integral will be represented by the equations (102), where 
x?, 2°, pt must be functions of n —1 independent variables, so that 
these equations represent, in fact, a manifold of m dimensions. But 
in addition these 27 +1 functions «,, z, p, of nm independent vari- 
ables must satisfy the relations 


(103) F(x, &, Pr = Fim; oN, Way @ Diy ke Sr 0, 
dz — p,dx, — p,dx, —-+--— p,dx, = 0. 


V, § 87] EQUATIONS OF THE FIRST ORDER 2638 


Since the differential equations (101) have the integrable combina- 
tion dF = 0, the first of the relations of (103) will surely be satisfied 
if F(a}, 2°, pt) = 9. On the other hand, we also have from the equa- 
tions (101) 


az day, dx 


dunt du ye woe 


n 
. 


du 


Since the initial values 2, 2°, pt are functions of n — 1 independ- 
ent variables v,, v,, +++, U,_1, we must also have 


U = 02 — p,dz,'— -.: — p, dx, = 0, 


where the letter 6 denotes the differentials corresponding to arbitrary 
increments 8v,,---, v,_, of these variables. By proceeding as in 
the case for n = 2, we have necessarily 


dU = déz — p,dédx, —---— p,d dx, — dp, dx, — --- — dp, dx,, 
dz=p,dx,+---+p,dx,, 
ddz =p, ddx,+---+p,ddx, + dp,dx,+.---+ 8p,dz,, 


and, since we may interchange the order of the operations d and 4, 
dU = 6p, dx; = dp; bx; 
Dy i 
= > §P,8p; + (X; + p,Z) 8x,3 du. 
te 


Since 2, %;, p, satisfy the equation F = 0, we have 


> (P, 8p; + X,8x,;) = — Z bz 


and, consequently, 
dU =— ZU du. 


From this we find the following expression for U: 
erg eae 


In order that U shall be zero, it is necessary and sufficient that U, be 
zero, that is, that we have 


62° — pi dat —---— pide? = 0. 


To sum up, in order that the equations. (102) represent an integral, 
it is necessary and sufficient that the initial values (a), 2°, pt) be 


264 PARTIAL DIFFERENTIAL EQUATIONS [V, § 87 


functions of n —1 independent variables satisfying identically the 
conditions 


(104) F(x}, x, Pi) a 0, 
(105) d2° — pda? — p§dag —.--— prdxo = 0. 


Every system of 2n+1 functions (a, 2°, pt) of »—1 variables 
satisfying these conditions defines an (m — 1)-dimensional manifold 
of elements. Again, we may say that every integral of the equation 
F = 0 is generated by the characteristic curves through the different 
elements of a manifold of this kind. 

In particular, to obtain Cauchy’s integral, which for x, = x} reduces 
to a given function ®(a,,---, x,), if we take x$, aj,---, 2) for inde- 
pendent variables (#} being supposed constant), the relation (105) 
gives the values of 2°, p$,---, °, 


O® o® 
o—@® sO cant 0 o = — aiene 0 — —. 
& (x3, Nery P2 Oxo’ » Pn O° 


The value of p} can be obtained from the relation (104). If P® is not 
zero (as we must assume in order to apply the general existence 
theorem of I, § 194, 2d ed.; § 188, Ist ed.), p? will be an analytic 
function of #3,---, #2 in a certain region, and the equations (102) 
will give, for 2, x,;, p,, analytic functions of wu, 23,---, 22. Moreover, 
the Jacobian Ce en 

D(u, Xo, sas Xn) 


is not zero, for it reduces to P$ for « = 0. Hence we can solve the 
first n equations (102) for wu, x},---, 2%, and, putting these expres- 
sions in the last of the equations (102), we obtain for z an analytic 
function of the variables x,, 7, +--+, 2p. 


Note. It may happen that the application of the preceding general rule does 
not lead to an integral. For example, it might turn out that the manifold of 
elements defined by the equations (102) does not really depend upon n arbitrary 
parameters. This is what would happen if the manifold formed by the elements 
(x?, 2°, py) were composed of characteristic strips; in this case, in fact, the 
manifold defined by the equations (102) would coincide with the manifold of 
the elements (2?, 2°, ph). 

Disregarding this case, it may also happen that the elimination of the param- 
eters U, Vy, °++, U—1 from the equations (102) leads to several distinct relations 
between the variables z,,--++, 2, Z. In order not to reject such solutions, we 
agree with Sophus Lie to enlarge the definition of the integral and to designate 
as an integral of the equation F = 0 every system of «” elements (x;, 2, px) Sat- 
isfying the relations 


(106) F (x, 2, pr) = 0, dz = p, dt, + +++ + Pndty. 


V, § 88] SIMULTANEOUS EQUATIONS 265 


IV. SIMULTANEOUS EQUATIONS * 


88. Linear homogeneous systems. Let us consider a system of 
q linear homogeneous equations in one unknown, f, 


: of 0 0 
RP)= ay Ge + ag ge +o + Ou ZH =O, 
Ee OF: meth Os Of) 3. 
(107) PCI IN ra OAc att a ah a 
0 0 7) 
XL) tae get Mag ge ts tag 5 = 0, 
2 n 


where the coefficients a,, are functions of the m independent variables 
Wy) Lay +++, Z, and do not contain the unknown function f. The ¢ 
equations (107) are said to be independent if there does not exist 
any identical relation of the form 


AAP) + +++ FAX AS) = 9, 


where A,, A,,---, A, are functions of x,, x,,--+, x, not all zero. It is 
clear that every system of g equations that are not independent can 
be replaced by a system of g’ independent equations (q'’< q) equiva- 
lent to the first, and that no system can contain more than n inde- 
pendent equations. 

We can therefore always suppose the g equations (107) inde- 
pendent and g =n. 

If g = n, and if the equations (107) are independent, the deter- 
minant of the coefficients a,, is not zero, and these equations have 
no other common integral than the trivial solution f= C, which we 
shall hereafter discard. If q is less than n, we can always find the 
integrals common to the equations (107) by successive integrations. 
In fact, let us suppose that we have integrated one of these equa- 
tions (the first, for example), and let y,, y,,---, y,—-1 be a system of 
nm — 1 independent integrals. Again, let y, be another function such 
that the Jacobian D(y,, ¥,, +--+; Yn)/D(@y %) +++) %) 18 not zero. 
Then we may take y,, y,,---, y, for new independent variables, and 
the equation X,(f)= 0 becomes @f/éy, = 0 by this change of vari- 
ables, while the equation X,;(f) = 0 (¢ >1) is replaced by an equation 


* We shall limit ourselves to an indication of the principal methods in their essen- 
tial features. For further details the reader is referred to E. Goursat, Sur Vinté- 
gration des équations aux dérivées partielles du premier ordre (Paris, Hermann 
et fils, 1892). 


266 PARTIAL DIFFERENTIAL EQUATIONS [V, § 88 


of the same form, in which the term in @f/éy, may be suppressed. 
This new equation may be written in the form 
of of 
ee ate 7h kar oa 0, 
where the coefficients 0,; are functions of y,, y,,---+, Y,- If we sup- 
pose the coefficients 6,; arranged according to powers of y,, this 
equation can be written in the form 


of ) 
Y; ; He e+ 6. 
(f)= («54 v Cn-~1, Oy, 4 


an rn( ge 

where the coefficients ¢,;, cj;,:-- are independent of y,. Since the 
unknown function / must be independent of y,, this function must 
satisfy all the linear equations which are obtained by equating to 
zero all the coefficients of the different powers of y,. Suppose that 
we proceed in this way with all the equations X,;(f)= 0 (¢>1). If 
the system formed by all the independent equations which we thus 
obtain contains m — 1 equations, the only solution is f= C. If not, 
the system will be composed of 7 linear independent equations 
(r<n—1). We may operate in the same way on an equation of the 
new system, and so on in the same manner. Since at each operation 
the number of independent variables is diminished by unity, it is 
easy to see that the given system has no other integral than f= C, 
or else it reduces to a system composed of a single linear equation. 

This method, which may be easily applied in certain cases, is 
evidently very imperfect from a theoretical point of view, since it 
does not enable us to determine a priori whether the equations 
(107) have common integrals other than f= C. We shall now show 
that this question can be settled without any integrations. 

Let f be an integral common to the equations (107). This func- 
tion satisfies the two relations X;(f) = 0, X,(f) = 0, where i and k 
are any two of the indices 1, 2,---, g. We also have 


XI%N]=*X0=0,  X,LX(f)] = %.(0) = 0 


and, consequently, 


Te tg )+ AO *)s 


Xi[X(P)] — 4) = 9. 
We have already observed that this new equation contains only deriva- 
tives of the first order (§ 36), and that it may be written in the form 


X[XP)] — GLX) = D3 (Xi (Gia) — X,(an)} 32 = 0. 


V, § 89] SIMULTANEOUS EQUATIONS 267 


Suppose that we form all the equations, similar to the preceding, 
obtained by combining any two of the given equations. These equa- 
tions have all the integrals of the system (107). Let us indicate by 


Xqii1(f) = 9, Xq+2(f) = 9, ey? Xq+s(f) = 0 
all those of these new equations which are independent of each 
other and which form with the equations (107) a system 


X,(f) = 9, ee X(f) = 9, 

ee eas = 0, 4 ae Xq+s(f) =9 
of independent equations. If g +s =n, the system (108), and con- 
sequently the system (107), has only the solution f=C. Ifg+s <n, 
we repeat on the system (108) the operations performed on the first 
system, and so on in the same manner. Continuing in this fashion, 
we finally obtain either a system of m independent equations, in 
which case the system (107) will have only the solution f= C, or 
else a system of 7 independent equations (7 < n) such that all the 
combinations X,[X,(f)]— X,[X;(/)] are linear combinations of 
X,(f), +++, X,(f). Such a system has been called by Clebsch a 
complete system. 

It follows, then, that the search for the integrals of a system of 
the form (107) leads to the integration of a complete system. 

Since it is clear that every system of n linear independent equa- 
tions is a complete system, we may say that every linear system 
reduces to a complete system. 


89. Complete systems. The theory of complete systems rests upon 
the following properties : 


1) Every complete system is transformed into a complete system 
by any change of variables. 


Let Cas Pi (Yr) Yor°' 56 Yn) (2 — a 2, mesy 2) 


be the formule that define a change of variables such that we can 
express also the variables y,, y,,---, y, 1n terms of the variables 
1, #,+++, %,- By means of such a transformation every symbol of 
the type af 


a ——s 
Ore 


7) 
X(f)= age tot 


where @,, @,++ +) Gs are functions of 4) Ly9***y) %,, Changes into an 
expression of the same form, Y(f)=0,0f/éy, +---+6,0f/0yn, 
where 3,, 6,,---, 6, are functions of y,,---, Y,. We have identically 


X(f)=Y(f), where the letter f on the left-hand side denotes any 


268 PARTIAL DIFFERENTIAL EQUATIONS [V, § 89 


function of x,, x, +++, x,, and on the right-hand side the same func- 
tion expressed in terms of the variables y,, y,,-++, Yn: 
Now let 


(109) XS/)= 0, «+, X(f)=0 
be a complete system. By means of such a transformation this 
system goes over into the system 


(110) ¥i(f)= 9, hee) Y,(f) = 9, 
where X;(f)= Y,(f) identically, with the understanding just men- 
tioned concerning the interpretation of f on the two sides. This new 
system is also a complete system. For, since we have identically 


X(A=ViS) MeA=¥VCS) 
for any function f, we also have 
XLX(f)] = YMA = KALYAN 
KL) = Fea ee 


and, consequently, 


X[M I — SLA] = YORI] - YL]. 


Since by hypothesis the a (109) is complete, we have for any 
two indices 7 and & 


X[Xi(S)] ce X,LX(S)] ie: XS) he ener A,X,(f). 


Hence, after the transformation, we have 


YIM AI - MLV MI=Man)+ > +++ 4), 
where Aj, ---, A, indicate the results obtained by replacing z,, x,, +++, 
2 ANY sie by their expressions in terms of y,,---, y,. The new 
system is heretare a complete system. 


2) Every system equivalent to a complete system is also a Cones 
system. 


A system of 7 linear homogeneous equations in ¢f/0z,, 


(109') VAG ad) alge VAS yea 
is said to be equivalent to the system (109) if we have » identities 
of the form 


Z (SF) = Aye Xa (f) + AnnXe (faa tA, (f); (k =1, 2,+++,7) 


where the coefficients 4, are functions of x,,a,,---+ x, whose deter- 


minant is not zero. In that case we can express X,(f),-+-, X,(f)_ 
linearly in terms of Z,(f), ---, Z,(f), and the name equivalent | 


V, § 89] SIMULTANEOUS EQUATIONS 269 


systems is self-explanatory. The difference 7,[7,(/)]— 2,[2,(S)] 
can now be written in the form 


San Saerwr]-Saor] Sawn] 


hence it is equal to a sum of terms of the form 


An An§ Xp[4XiS)]— XS) § + An Xn An) XP) — AuX( An) Xf): 
If the system (109) is complete, this difference will therefore be a 
linear function of X,(f),---, X,(f), since all the differences 


XX) - OL) 
are, by hypothesis, linear functions of X,(f),---,X,(f). Since the 
two systems (109) and (109) are equivalent, all the differences 


2;[2.P)]— 4[2:7)] 
can be expressed linearly in terms of 7,(f), Z7,(f), +++, Z,(/). 

It is clear that every complete system can be replaced by an 
equivalent system in an infinite number of ways. We say that the 
complete system (109) is a Jacobian system if all the expressions 
X,;[X,(f)] — X,[X;(f)] are identically zero. We shall now show 
that every complete system is equivalent to a Jacobian system. 

Since the r equations (109) are independent by hypothesis, we 
can solve them for 7 of the derivatives of f, for example, for the 
derivatives @f/0x,,-+-, 0f/ex,. Since the system thus obtained, 


ge a Oe OF, 
AS ie cas a mS are Scar snr Be = 0, 
ge of af 
airy | AU iat tate gt Tene, = 
Se ee ee Le er eis be te WOR oS or) 
7) 9 3 
z,(f)= sE + elas Oy, 9 ake 0, 


is equivalent to the system (109), it also is a complete system. Now 
if we form the expressions Z7,[Z,(f)]—2,[2;(/)], it is clear that 
only the derivatives 0f/0x,,,, +++, Of/ex, will appear, and conse- 
quently the new equations 7,;[2Z,(f)]— 2,[2:(f)]=9 can be 
linear combinations of the equations (111) only if the left-hand sides 
of these new equations are identically zero. The system (111) 1s 
therefore a Jacobian system. 

The reasoning proves that every complete system of the special 
form (111) is a Jacobian system, but it is clear that a Jacobian sys- 
tem is not necessarily of that form. 


270 PARTIAL DIFFERENTIAL EQUATIONS [V, § 89 


3) Every complete system of r equations in n independent variables 
can be reduced by the integration of one of the equations of the system 
to a complete system of r — 1 equations in n — 1 independent variables. 


Suppose that we have integrated one of the equations of the sys- 
tem, for example, the equation X,(/)= 0, and that we choose a new 
system of independent variables (y,, y,, +--+, Yn), a8 in the preceding 
paragraph, in such a way that y,, y,,---, y, are n —1 integrals of 
X(f)= 9. The system (109) is replaced by a new complete system 
in which the first equation reduces to éf/éy, = 0. Solving the r—1 
remaining equations for the » — 1 derivatives éf/dy,, ---, Of /ey,, for 
example, we obtain a complete system, 


V(f) = 52 = 0 


ene We Ge 
(112) Ledges = a Coy OY, 44 Tartans "a 0, 


af af af 
te —_ ji + C,. Gases ye -f- eee + Ca Pees Ue — @) 
(7) OY; ’ OY +1 ‘ OYn ‘ 


which is of the special form (111) and which is therefore a J acobian 
system. Now we have 
ar do, nn Of 


ELPA GO Ped AL GO] be ie Oy, 44 i oy, oy, 


and since this expression must be identically zero, we see that the 
coefficients c, of the new system are independent of the variable y,. 
Moreover, for 7 >1, k >1 we have identically 


YiL¥.S)] — YLVi)] = 93 


consequently the r — 1 equations 


(113) Y¥,(f)= 9, Y,(f)= 9, ee) Y,(f)=0. 
form a Jacobian system of r—1 equations in » —1 independent 
variables ¥,, Y%, +++) Yn» Which establishes the proposition. 

The system (113) can in turn be reduced to a complete system of 
r — 2 equations in m — 2 independent variables, and so on in this 
way. Continuing in this manner, we finally reduce the given com- 
plete system to one linear equation in n — 7+1 independent vari- 
ables. We conclude from this that every complete system of r equations 
in n independent variables has n — r independent integrals, and the 
general integral of the system is an arbitrary function of these n —r 
particular integrals. 


V, § 89] SIMULTANEOUS EQUATIONS OTL 


The preceding reasoning shows also what are the integrations to 
be carried out in order to obtain these integrals. Moreover, it is clear 
that this method can be applied in a variety of ways. We may, in fact, 
replace the given complete system by any other equivalent system, 
and begin by integrating any one of the equations of this new sys- 
tem. For example, if we replace the complete system by a Jacobian 
system of the form (111), we know at once rv — 1 particular integrals 
z,,++-+, 2, of the equation Z,(f)= 0, and it is sufficient to integrate 
a system of n —,r ordinary differential equations in order to have 
the general.integral. For complete details of other methods of inte- 
gration of complete systems, the reader is referred to special treatises. 


Example. Let it be required to integrate the system 


aT) a CET TC cia ana 0, 
(114) . B 


Forming the combination X,[X,(f)]— X,[X,(/)], we are led to add to the 
given equations a new equation 0f/0z2, + 2, 6f/éx, = 0, and the system of three 
equations thus obtained is equivalent to the system 


of o Of _ of of of Ol, 
(115) Fe abe e, 0 eee or: a (0h Pa CES: aati 
which is a Jacobian system. The system (114) has therefore only one independ- 
ent integral. The general integral of the last equation of this system is an arbi- 
trary function of z,, 7,, and 7,—2,2,. If we take for independent variables 
Ly,.%, Tz, and u= 2, —2,7,, every function f(z,, 2, %,, ©,) changes into a 
corresponding function ¢(z,, 22, 7,, u), and the system (115) is replaced by the 
system 


(116) BO ie eee Oa, Ss jn OF Sg, 
OX, u 


The first two equations of (116) form a new Jacobian system of two equations in 
three independent variables z,, x, u. The general integral of the second is an 
arbitrary function of x, and of u — «3/2. 

Let us now take for independent variables 7,, z,, and u — ta / ies, Mvery 
function $ (2, 2, u) changes into a corresponding function y (a, x, v), and 
the first two equations of (116) become 


rest) 
OL, 
The general integral of the first is an arbitrary function of »v — 29, and, conse- 


quently, returning to the original variables, we see that the general integral of 
the system (114) is an arbitrary function of 


inate 


272 PARTIAL DIFFERENTIAL EQUATIONS [V, § 90 


90. Generalization of the matey of the complete integrals. Let us 
consider an equation 


(117) V (Wy Wgy ++ Uyy %5 Uys Ugy ++) Mq_y gi) = 0 


defining a function z of the m independent variables x,, x,,---, x, 
which depends also upon (n —7 +1) arbitrary parameters a,, a,, 
++, @,_,41- If we suppose that definite values have been assigned 
to these parameters, and if we eliminate them from the relation 
(117) and the relations obtained by successive differentiations, 


OVS OV dz 


(118) —— + p,=— = 0, D=-a—? (4 =1, 2,-+-, 2) 


we obtain in general only r independent relations between z, x,,---, 
Hay Py * ty Pns 
(119) F,(@,, +++, Gq, 2% Diy s+) Pa) = 9, F,=90, ---) FF = 0. 


Limiting ourselves to this case, which is the general case, we shall 
say, as above (§ 82), that the function z defined by the relation (117) 
is a complete integral of the system of partial differential equations 
(119). We shall show that, in this case also, the knowledge of a 
complete integral of the system (119) enables us to find all other 
integrals. In fact, since the equations (119) arise from the elimina- 
tion of a,, @,,+++, G—,41 between the equations (117) and (118), 
finding an integral common to these 7 equations (119) reduces to 
finding a system of functions 2, a,,---, @_,4, of the variables x 
H,++*, , Satisfying the equations (117) and (118). It is obvious 
that we can replace the system of equations (117) and (118) by the 
system consisting of the equation (117) and the equation | 


(120) ft +55, Ms +...+ 


© 


pate Ady p41 oH 0, 
which is obtained by HEY et the equation (117) and making 
use of the equations (118). We can ey the equations (117) and 
(120) in a variety of ways: 
1) By supposing that a,, a,, - - 
precisely the complete integral. 
2) By putting 


+, @,_»41 are constants, which gives 


OV OV 


Y= 0, ado. cle ey = 0), 


OO 


The elimination of a,, a,,---+, a,_,41 from these equations, if it is 
possible, furnishes an integral which does not contain any arbitrary 
constant, and which we shall call, as before, a singular integral. 


V, § 90] SIMULTANEOUS EQUATIONS 273 


3) If all the coefficients ¢V/éa; are not zero simultaneously, there 
exists at least one relation between the unknown functions a,, a,, 
+5 Q,_,-41 Of the variables x, (I, § 55, 2d ed.; § 28, 1st oat 
Suppose that there exist k and only & independent Helaticn between 
these functions, 


(121) Fey 2) Oy 41) = 9, ss "9 Te (@y Uy) +++) An_ pai )= 0 


Since the relation (120) must be a consequence of the relations 
df; = 0(i =1, 2,---, k), there exist k coefficients ,, A,,---, A, such 
that we have identically 


é OV 


TL Sree Pe 


5g, it be 


Dos eg = hh ns A OF 


This eee is equivalent to n — 7 + 1 distinct relations, 


ae th OF, Of: 
piaet coger Ve aakoa 
(122) rege. Spat LITE i ES? J AAA ig 
Aver res Cr Of; 
@0y pai ie a O6y pt rf iy sh Me On Ses 
The elimination of @,, @,, +++, G_—»+1) Ay Ay + +» Ay from the equations 


(117), (121), and (122) will lead, in general, to a single relation 
between 2,, @,,-+-, x,, and z, that is, to an integral common to the 
equations (119), which depends upon the arbitrary functions chosen. 


_ The set of integrals thus obtained, by making the number & vary 


from 1 to n —r, and by taking the functions f, f,,---, f, arbitra- 


_vily, constitutes the general integral of the system (119). It will 


be observed that the complete integral will be obtained by supe as 
k=n—r-+1. 

iit 1, the system (119) reduces to a single equation. Con- 
versely, given any equation of the first order F(a, 2; p,)= 0, it 
follows from the general existence theorems that it always has an 
infinite number of integrals which depend upon as many arbitrary 
parameters as we wish, and consequently an infinite number of com- 
plete integrals. The preceding method, which is a direct generaliza- 
tion of that of § 82, enables us to find all the other integrals of the 
equation / = 0 when we know one complete integral. 

If ry >1, the system (119) is not the most general of its kind, for 
a system of 7 equations of the first order with a single dependent 
variable does not necessarily have any integrals. We shall show in 
the following paragraphs how to determine whether such a system 
is consistent, and how to find the integrals when they exist. 


274 PARTIAL DIFFERENTIAL EQUATIONS [V, § 91 


91. Involutory systems. Let 
(123) F (2, Bay %**y Loy A ear ed NW F, = 0, reece Pet 


be a system of r independent partial differential equations of the 
first order, not containing the dependent variable z. The general 
case can always be reduced to this particular case by the device 
used in § 75. The problem of finding an integral common to the r 
equations (123) is equivalent to the following problem: To find n 


functions p; = $;(#, +++, ®,) satisfying the relations (123) and the 
conditions Op,/0x, = Op,/Cx;. 
If we know a system of n functions ¢;(#,, -- +, ,) satisfying these 


conditions, we can derive from them, by quadratures, an integral of 
the equations (123) which depends upon an arbitrary constant. 

Let F and H be any two functions of the 2m variables ~,, p,. 
Using the notation (see § 81) 


(F, 1) = SG eee 


we shall call the expression (/’, H) a Poisson parenthesis. We now 
have the following theorem: Jf the two equations F = 0, H = 0 have 
a common integral, that integral also satisfies the equation (F, 7)= 0. 

For let us suppose that p,, p,,---, p, are functions of the n vari- 
ables #,,---, , satisfying the two equations F = 0, H = 0 and the 
conditions ép,/0x, = Cp,/ex,. Differentiating the relation F = 0 with 
respect to x;, we find 


Bip ss AEF 
0x; > dp, Ox, 


Multiplying this equation by 0H/dp; and adding all the similar 
resulting equations, we find 


H OF Op, _ 


=), 
a 0x, Op; fs ON; op, Ox; 


») 


Permuting the letters F and H and observing that we may permute 
the indices 7 and & in the double sum, we have also 


LOH OF (it te OF OH Op, 
5) Ox, Op; ap ps Op, Op; Ox, 


Ved ka=1 


Subtracting the two results term by term, it follows that 


OF 0H (0p; Op, 
(124 F, H)+ > Se Ee ae) = Q, 
We a1 ta Pa Op; \Oay, Oar, 


V, § 91] SIMULTANEOUS EQUATIONS OTe 


If p,,--+, p, are the partial derivatives of the same function, we 
have, for any two indices ¢ and k, 0p,/0a, = Cp,/éx; and, consequently, 
CREA 0, 

This theorem contains as a particular case the one which was 
proved above (§ 88) for linear homogeneous equations in p,, +++, Pay 
and its logical consequences are also analogous to those of § 88. For 
every integral of the equations (123) is also an integral of all the 
equations (F,, Fg)= 0 which can be formed from pairs of the equa- 
tions (123). Hence we can adjoin to the given system all of these new 
equations which form with the original equations a system of inde- 
pendent equations. Continuing in this way, we must finally obtain 
either a system of independent equations whose number exceeds n, 
in which case the system has no integral in general, or else a system 
of m equations (m = n) such that all the equations (F,, Fg) = 0 are 
satisfied identically or are algebraic consequences of the preceding. 

Such systems are similar to complete systems. It is always possi- 
ble either to show that the given equations are inconsistent or to 
reduce them to a system for which all the parentheses (F,, Fg) are 
identically zero. In fact, let us suppose that we have solved the r 
equations (123) for r of the variables p,,--+-, p,, which must always 
be possible, for otherwise the elimination of p,, ---, p, from these r 
equations would lead to a relation between the variables 2,,---, x,, 
and the given system would evidently be inconsistent. Let 


(125) Py fi (Pras ne a Se) HX) ie os L,) = 0, RE: eG r = 0 
be the equivalent system thus obtained. The parenthesis 


(De ey Pp A) 


does not contain any of the variables p,,---, p,; hence the equations 
obtained by equating these parentheses to zero cannot be conse- 
quences of the first, and they furnish new equations if the paren- 
theses are not. identically zero. Solving these new equations for 
certain of the quantities p,.,,-+-+,,, and continuing in the same 
way, we finally either demonstrate the impossibility of the problem 
or else obtain a system of m equations of the first order (m = n), 


(126) Fis Ope 8a Fee, 


such that all the parentheses (7, Fg) are identically zero. 

Such systems, which are similar to the linear Jacobian systems, 
are called involutory systems. It follows from what precedes that 
the search for the common integrals of a system of equations of the 
first order reduces to the integration of an involutory system, 


276 PARTIAL DIFFERENTIAL EQUATIONS [V, § 91 


This integration is immediate if m = n, as the following proposi- 
tion shows: Let F,, F,,--+-, F, be functions of the 2n variables x;, 
Pry such that all the parentheses (F,, Fg) are identically zero, and 
such that the Jacobian A = D(F,,+++, Fy)/D( Py +++) Pn) US Not zero. 
If we solve the n equations 


(127) Fi, = 4) Fy = G, tesy i =="0., 


Where A,, Ay +++) My, are any constants, TOP Ds Daromte Pas the expres- 
sion p,dx,+-+-++p,dx, is an exact differential for the resulting 
values of the p’s. 


For we have se... 
(Fa — Gay Fy — %g) = (Fas Fp) = 0 


and, by what precedes, these n functions p,, p,,---, p, of the n varia- 
bles #,,#,,-+-+,#, defined by the n equations (127) must satisfy all the 
relations 


phe dat ty OF, OF, Ope Op; si 
>> 2 api seats on) = 0. (a, Bol 2, vey nN) 


Let us take all the n relations of this kind in which the index B 
retains the same value. These relations can be written in the form 


“. OF, wi OF, (Op On; 
a irs) k 
>> OD; > OD, (s ae Me 


If we take for unknowns the nm expressions 


OF, [0 j ‘ 
SESS on (i =1, 2,---,n) 


Opry On, 
the determinant of the coefficients of these unknowns is precisely 


the determinant A, which, by hypothesis, does not vanish identically. - 
It follows that we have, for any two indices 7 and B£, 


> OF, oe sl a) a 

mI Op, \ Ox, On, 

Similarly, taking the n equations of this kind in which the index i 
has a definite value, we evidently have ¢@p;/éx, = 0p,/ex;, which 
proves the proposition. 


The function 
Ata b(x,, tty Lay A) Mk On+1) 


128 
( ) =f (rydey bo FP) + aug 


where a,,, 18 a new arbitrary constant, represents the complete 
integral of the involutory system (127). If we regard the 7 con- 
stants a@,, a,,+-+-+, a as having definite values, while the constants 


V, § 92] SIMULTANEOUS EQUATIONS OTT 


G41) ***, G4  Yemain arbitrary, the formula (128) represents a 
complete integral of the involutory system formed from the first r 
equations of (127). This is a true complete integral, for, from the 
way in which we have obtained the function #, the equations 
o® O® 
Px at pies Se ? Pn car ox, 


1 
form a system equivalent to the system (127), and the only inde- 
pendent relations not containing a,,,,---+, a, which can be deduced 
from them are evidently the first r equations of this system. 


92. Jacobi’s method. Let us consider an involutory system of r 
equations (r < n), 


(129) F,(@,, s 2% Xn5 Py,°° *» Pn) aed § YA +) Bey, ay "> Pa) = a, 
where the constants a,,---, a, have definite values. To obtain a 
complete integral of this system, it is sufficient to adjoin to itn —r 
new functions F,,,,---, F,, such that the Jacobian 


n?) 


DLE. resy F,) 

Da she 3 Pn) 

is not zero, and such that the new system 
(130) Fy = Ayes, F,, = 4,, Fn 1 = Oy 415 eval Fi, = 


is itself involutory. Indeed, the complete integral of this system (130) 
will furnish, as we have just seen, a complete integral of the system 
(129). If r=1, this method is merely the extension of the method 
of Lagrange and Charpit to an equation in m variables. 

Jacobi’s method for solving this problem depends upon a noted 
identity due to Poisson. Let f, 4, » be any three functions of the 
2n variables x,, p,; then we have identically the relation 


(131) CS p); Wy) an ((4, W); I) = i ((y, Ds p) = 0. 


In fact, each term on the left-hand side is the product of a partial 
derivative of the second order and two partial derivatives of the 
first order. Hence, to show that it vanishes, it is sufficient to show 
that it does not contain any derivative of the second order of the 
function f, for example, since the three functions jf, ¢, y appear in 
it symmetrically. The terms containing the second derivatives of f 
can arise only from ((f, $), ¥) + (sf); 6) = Ws (#f)) — (# Ws A))- 

Observing that (¢, f) and (y, f) are two linear homogeneous expres- 
sions in the derivatives of f, and setting 


PAHS)  (HL= PS): 


278 PARTIAL DIFFERENTIAL EQUATIONS [V, § 92 


the preceding expression can be written in the form 


Y[X(f)]-ALYO)]- 
Now we saw in § 88 that this expression does not contain any second 
derivatives of f. It follows that all the terms of the left-hand side 
of the equation (131) cancel each other in pairs. 
Finally, in order to integrate the involutory system (129), let us 
first try to find a function ® independent of F,,---, F, satisfying 
r linear homogeneous partial differential equations of the first order, 


(132) (Ff, ®)=9, (F,, &) = 0, oo (FO) =U 
These r equations form a Jacobian system. For if we set 


: j f X;(®) = (Fi, ®), 
Poisson’s identity, 


((Fa, Fe), ®)+((Fp, ®), Fa) +((®, Fa), Fe) = 0, 


Xa[Xp()]—Xe[X.(®)] = 9, 
S1nGe) (ha, 2a =U: 
Let F.,, be an integral of this Jacobian system which forms with 
F,, +++, F, a system of independent functions of p,,---, p, We 
next proceed to form the new Jacobian system of 7 +1 equations, 


as d)= 0, he aD (Fa 6) —:0; 
and to find an integral of this system which is independent of 
F.,-++, #4, a8 functions of the »;; and we continue in the same 


way. Finally, when we have found an integral of the last Jacobian 
system, 


becomes 


(F,, &)= 0, 2 89 AF, a1) = 0; 


n 


we can obtain a complete integral of the given system by a quadra- 
ture, aS we have seen above. 


V. GENERALITIES ON THE EQUATIONS OF HIGHER ORDER 


93. Elimination of arbitrary functions. The study of partial differ- 
ential equations of the first order in a single dependent variable has 
led us to the following conclusions: 1) The integration of an equa- 
tion of this form reduces to the integration of a system of ordi- 
nary differential equations. 2) All the integrals of this equation are 
represented by one or more systems of equations in which appear 
explicitly one or more arbitrary functions and their derivatives. 

These properties are not extensible to the most general partial 
differential equations of order higher than the first. The problem 


V, § 93] EQUATIONS OF HIGHER ORDER 279 


of the integration of such an equation cannot, in general, be reduced 
to the integration of a system of ordinary differential equations. 

We can easily generalize, however, the method of the elimination 
of arbitrary functions which leads to a partial differential equation 
of the first order ($§ 77 and 82), but the equations of higher order 
which we obtain in this way form only a very special class. 

Thus we have seen that the general integral of a linear equation 
in two independent variables Pp + Qq = R is obtained by associating 
the curves of a congruence according to an arbitrary law. Let us 
now consider a family of curves T that depends upon n + 1 arbitrary 
parameters @,, @,, +++, G41, (n >1), 


(133) F(a, Y; % Gy ++) n41) =9, PCH, Y, % Ay, +++) My 41) = 9. 


If we establish n relations of arbitrary form between these n +1 
parameters, we obtain a family of curves I that depends upon only 
one parameter. These curves generate a surface S, and all these 
surfaces S satisfy, whatever may be the m relations established 
between the n + 1 parameters, a partial differential equation of the 
nth order, which is called the partial differential equation of the 
family of surfaces S. To prove this, let us observe that instead of 
establishing n relations between the n +1 parameters a,, it amounts 
to the same thing to take for these parameters arbitrary functions 
a,(X) of an auxiliary variable X. The two equations (133) then define 
two implicit functions z= /f(#, vy), X= ¢(«, y), and we have to 
prove that the function z=/(a, y) satisfies a partial differential 
equation of the nth order, independent of the form of the arbitrary 
functions a;(A). Differentiating the first of the equations (133) with 
respect to « and then with respect to y, we obtain the two relations 


OF OF OF , Ole iy pie 
Get Get |p WAT te win) |X =O, 
OF OF Ona: OPEN Hod 
Gy tet t [an BOF +3 nO) |X = 0, 
from which we derive 
hi Oi 
Deguney ti ddesas 
x, OF, OF 
Ox pa 


From the second of the equations (133) we derive, similarly, an 
expression for the quotient 7/X;, which is deduced from the pre- 
ceding by replacing in it F by ®. Equating these two expressions, 


280 PARTIAL DIFFERENTIAL EQUATIONS [V, § 93 


we adjoin to the equations (133) a new equation containing 2, y, z, 


Ps Ws U9 %qr* * *y Unt 
(134) Ws (2, Ys % Py Vy Uy My ++ *y ng) = 9. 


Operating on this new equation as on the equation ' = 0, we derive 
from it an expression for d//A;, which depends upon @, y, 2, p, 9, 7, 
S, t, @,)+++; 41 Equating this new expression to one of the expres- 
sions already obtained for this same quotient, we obtain a new 
relation containing the second derivatives of z, 


(135) W, (2, Ys; % Py 4,7, 8, t, Ay) By 2 *y Unt 1) = 0. 


After similar operations we adjoin to the system (133) a system 
of n relations containing @,, @,, +++, @,41, X, y, 2, and the derivatives 
of z up to those of the nth order. The elimination of @,, @,, +++, G44 
from these n + 2 equations will lead, in general, to one and only one 
equation between a, y, 2, and the partial derivatives of z up to those 
of the nth order. This is the partial differential equation of the 
surfaces generated by the curves I. 


Example 1. If the curves I are the straight lines parallel to the zy-plane, 
the equations (133) are 
Rix Oy, Y = 4,0 + ag. 


Applying the general method, let us suppose that a,, a,, a, are functions of a 
parameter }. From the two preceding equations we derive for the quotient 
dZ/ the two values p/qg and — a,, which leads to the relation p/q¢ + a, = 0. 

Differentiating this last relation with respect to z and then with respect to 
y, and dividing the corresponding sides of the resulting equations, we find 


Equating this value of the quotient to the expression g/p already obtained, we 
find again the partial differential equation of the ruled surfaces which have the 
xy-plane for the directing plane (I, Ex. § 39, 2d ed.; § 24, Ist ed.), 


g?r — 2pqs + p?t = 0. 


Example 2. If the curves © are all possible straight lines, the equations (133) 
can be written in the form 


C=A2+4, Yroazet a. 


Applying the general method, we derive from them successively 


Ny ae 1 gs 
Ny 4p-l Asp s 


or 


V, § 93] EQUATIONS OF HIGHER ORDER 28) 


From this new equation we then derive 


AVE G8 Gat) | Gy 
Stic > Sans HANMER 
N, r+ 43s As 
or 
(A) air + 2a,a,8 + azt=0. 
This last equation gives in turn 
My GPa + 20,4,P ig + UGPog ieee Gthz 
eige shat x Se Raa, URES paar tree >? 
Nz YDgq + 24,43 P2 + O3P yp ag ex Oy" 
or, clearing of fractions, 
(B) G{Dyq + 8 azpaz Py, + 34, A3D1_ + A3Pog = O. 


Eliminating the quotient a,/a, from the relations (A) and (B), we obtain the 
partial differential equation of all ruled surfaces. We see that this equation 
contains only derivatives of the second and third orders. By its derivation 
we see that it states that at each point of the surface one of the asymptotic 
tangents has contact of the third order with the surface (I, § 223, 2d ed.; 
§ 238, 1st ed.). 

Example 3. Let us consider the plane curves [ represented by the two 


equations 
; z=f (x, Y; iy, °**) Gn), Y= Q+1- 


Instead of applying the general method, let us suppose that a,, a,, ie +, Gy, are 
functions of the last parameter a,41. The surface S generated by these curves 
I’ has for its equation 

: z= S[@,Y, dy(Y)s-++1 Gn) ]s 


where ¢,, +--+, ¢n are arbitrary functions of y. The elimination of these n func- 
tions from the preceding relation and the relations which give 0z/éx, 0?z/dx?, 
+++, 0"z/dz" leads to a partial differential equation of the nth order, 


onz 02 on—1z 
(136) —= F(2, Ys ise oe —"): 


in which only derivatives with respect to z appear. 

Conversely, every partial differential equation of this type can be integrated 
as an ordinary differential equation containing a parameter. If z=f(a, y, Cj, 
C,, +++, Cn) is the general integral of such a differential equation, it will suffice 
to replace C,, C,,---, C, in it by arbitrary functions of y in order to have the 
general integral of the same equation, considered as a partial differential equa- 
tion in two independent variables x and y. 


The general integral of a partial differential equation of the first 
order, of any form, in two independent variables, is obtained by tak- 
ing the envelope of a two-parameter family of surfaces when we 
establish an arbitrary relation between these two parameters (§ 82). 
To generalize this result, let us consider a family of surfaces & 
which depends upon n + 1 parameters, 


(137) F(a, y, 2; @,, Uy" **y On 41) = 0. (n > 1) 


282 PARTIAL DIFFERENTIAL EQUATIONS [V, § 93 


If we establish n arbitrary relations between these n + 1 parame- 
ters, or, what amounts to the same thing, if we replace a,, @,,---, 
a,4, by arbitrary functions of an auxiliary variable A, we have a 
family of surfaces % which depends upon a single parameter. The 
envelope of this family of surfaces is a surface S which satisfies a 
partial differential equation of the nth order, independent of the 
form of the arbitrary functions a,;(A). For we should obtain the 
equation of this surface by eliminating A from the two equations 
(137) and (138) 


ar, as aaa 
(138) 8a, ay (A) +.--+ FRAO an (Vf 


But these two equations may be considered as defining two func- 
tions z= f(a, y) and X = ¢(a, ¥) of the two variables x and y. The 
partial derivatives p and qg are given by the two equations (I, § 41, 
2d ed.; § 25, 1st ed.) 


Ober OF 0H 
(139) mei ds Oie 0, =e 
Applying to this system (139) the method applied to the system 
(133), we can adjoin to it, step by step, m — 1 new relations between 
iy Uy °° *y Unaiy ©, Y, % and the partial derivatives of z of orders 
2, 3,---,. The elimination of a,, a,,---, @,4, from these n —1 
equations and the equations (137) and (189) will lead, in general, to 
a single relation independent of a@,, @,,---+, @4,, in which will 
appear x, y, 2, and the partial derivatives of z up to those of the 
nth order. 


Example. If the surface = is a plane, we find again the equation of the 
developable surfaces s? — rt = 0. If the surface = is a sphere with the constant 
radius R, the equations (187) and (139) become 

(t— a4)? + (y — ay)? + (2 — a4)? — B = 0, 
(140) 
%— a, + (2—a,)p = 0, y— a, + (2—a,)q = 0. 
Suppose that a,, a,, a, are functions of a parameter >. Equating the values 
of the quotient \//X;, derived from the last two equations (140), we obtain the 
relation 


(141) (rt—s?) (2— ag)? + (1+ p?)t+ (1+ 9’) r— 2 pqs] (2— as) +1+p?4+9?=0. 


We shall obtain the desired equation by eliminating a1, a2, a3 from (140) and 
(141). From the first we derive z — a, = R/V1+ p? + @, and, replacing z — dg 
by this value in (141), we obtain the partial differential equation of the tubular 
surfaces, 


(142) (rt — s?) R24 [(14p*)t+ (1+9?)r—2 pgs] R-V14-p?+ 2+ (1+p2+¢2)2=0. 


V, § 94] EQUATIONS OF HIGHER ORDER 283 


The geometric meaning of this equation is easily verified. It states that 
one of the principal radii of curvature of the surface is equal to R (I, § 242, 
2d ed. ; § 241, Ist ed.). 


Note. Given a function of several variables which depends upon 
one or more arbitrary functions, it is not always possible, as in the 
two cases which have just been examined, to deduce from them 
one and only one relation, independent of the form of the arbitrary 
functions, between the independent variables, the function z and 
its partial derivatives up to a given order. Let us consider, for 
example, a function z = F(a, y, X, Y), where F is a given func- 
tion of the four arguments which appear in it, and where X and 
Y are arbitrary functions of the variables x and y respectively. The 
five derivatives p, g, 7, s, ¢ of the first and second orders depend 
upon X, X', X", Y, Y', Y", and it is in general impossible to elimi- 
nate these six quantities from the six equations. But if we continue 
up to derivatives of the third order, we have, in all, ten relations 
containing eight arbitrary quantities, X, X', X", X'', Y, Y', Y", Y"", 
and the elimination will lead to a system of two equations of the 
third order.* 


94. General existence theorem. The proof given for a system of 
partial differential equations of the first order (§ 25) can be extended 
readily to the most general systems of the normal form, studied by 
Madame Kovalevsky, t 


O12 

Le, 3 
Ox" F(@; Hay *%y Uns Ky Ky 2%) Mpa?’ ), 
O22, 


(143) Oats = F (2; Ly) aes +f Oa @1) ®o) oi , Zn aia ), 


De e 
Out F (5 Ho» sty Las 1) %o) grees Rp) ay ‘); 


in which the right-hand sides contain the independent variables z,, 
@, +++, £,, the dependent functions z,, ---,2,, the partial derivatives 
of z, up to and including those of order 7,, the partial derivatives of 
z, of orders up to and including those of order 7,, ---, and so on, 


* See HERMITE, Cours d’ Analyse, pp. 215-229. 

+ Journal de Crelle, Vol. LXXX. In her proof, Madame Kovalevsky reduces the 
general case to the case of a linear system of the first order, but for us it will be 
sufficient to reduce the general case to the case of a system of the first order of any 
form whatever. 


284 PARTIAL DIFFERENTIAL EQUATIONS [V,.§ 94 


but none of the derivatives 0%12%,/0x}1, 0"2z,/Oxis, ---, Oz, /Oxjr. We 
may then state the general theorem as follows : 


Regarding the quantitres ©,, Ly, +++) Lay Z yy Zyy* + *y Bpy 
OM tag t ersten ye 
pride SAAS Sc Lhe 2 
Ons Onss » » « Oxen 

which appear in the functions F, as independent variables, let 


t 
ay) ay, ee ) Ans b; b,; ne *9 Bay Oe ates 


be any system of values of these variables in whose neighborhood the 
functions F, are analytic. On the other hand, let 


di, 1, Pi * +, PD, 
(144) do, $2; $3, ers —', 
Pps Pp» $35 eas eae 
be functions of the n—1 variables x, x,,-++, £,, regular in the 
neighborhood of the point a,,+++, A, and such that we have 


A+++ +a a: 
0%. bet ; 


Onks Sr Caren Ty gy 29+ By 


$; oz b;, 


for ©, = A+++, %, = a,. Then the equations (143) have one and 
only one system of integrals, analytic in the neighborhood of the point 
(Ay) Uy + ++, &,), and such that we have, for x, = 4,, 


Cz; ori-ly. 


i d;; Pay iy tes, PE eh Ts G@= 1; 2, s++, Dp) 


To prove this we observe first that the equations (143), and those 
which we obtain from them by successive differentiations, enable us 
to express all the partial derivatives of the dependent variables in 
terms of the independent variables, the dependent variables, and the 
partial derivatives 0%+-: +¢»z;,/dxt ..- Oxan, where a, < r, for 1 =1, 
a,<yr, fori =2,---,a,<7, fori=p. This follows, step by step, 
by a process of reasoning exactly like that of § 25. Now the ini- 
tial conditions determine immediately for x,=a,, ---, %, =a, the 
numerical values of the derivatives in terms of which all the 
others are expressible. Hence the coefficients of the developments 
in power series of the integrals whose existence we wish to prove 
can be calculated by the operations of addition and multiplication 
alone, in terms of the coefficients of the developments of the func- 
tions F; and of the functions of the array (144). 


V, § 94] EQUATIONS OF HIGHER ORDER 285 


To finish the proof, it remains to éstablish the convergence of the 
power series thus obtained when the absolute values of the differ- 
ences #;— a@,; are sufficiently small. We have already proved this 
convergence when all the numbers 7,, 7,,---, 7, are equal to unity. 
We shall now show how to reduce the general case to this particular 
case by considering as dependent variables the functions z,,---, z,, 
and their partial derivatives up to those of order r;—1, inclusive, 
for 2;(¢ = 1, 2,- +, p). 


Let us put 
Oty testo t en 


Dats Danka ©» » Dante Pe eyo (20, 0,-..,0 = %) 
The right-hand sides of the equations (143) contain the variables 
@4, +++, ®,, the dependent variables z,,---, 2,, the new dependent 
variables, and certain derivatives of the first order of these new 
dependent variables. But, by hypothesis, the derivatives of the varia- 
ble z; of order 7; which can appear are different from the derivative 
#..0,0,--,0- Hence at least one of the numbers @,, a,,---, a, is differ- 
ent from zero. If, for example, a, > 0, we can replace Ree, CO abo: 


Oz, GB — 1, Aye, @, 
0x, 
when a,+a,-+---+a,=7;, and similarly for the others. We can 


therefore write the given equations (143) in the equivalent form 


O Bim 1)05 0) 6: 
(145) Stata! aie or ®;(z,, Seba tn 5! Saye 2 9 Aig 5", t ); (@=1, 2,°° *) DP) 
1 
the right-hand sides containing only the independent variables and 
the dependent variables with some of the partial derivatives of the 
first order taken with respect to one of the variables x,,---, 2,. To 
these equations must be adjoined those which give the derivatives 
with respect to #, of the new dependent variables, other than those 
which we have already written. If we havea,+a,+-+---+4,37;— 2, 
we can write immediately 
oz 


B11 Foy 22 29 An seat ao 
(146) Ft Beare nia, the 


1 


pot a,? 


and we have a, +1+a,+---+a,=7;—1, so that the right-hand 


t 


side is one of the dependent variables. If we have 


G+ $a=%—1, 


286 PARTIAL DIFFERENTIAL EQUATIONS [V, § 94 


we must suppose a, < 7; — 1, and, consequently, one at least of the 


numbers @,,---, a, is different from zero. If, for example, we have 
a, > 0, we shall write 
(147) Oza Socenntn OR 1, Gy—1, ++ Ay 
ES SL poe Shed a ie TE 
Ox, 0x, 


and the right-hand side is the derivative with respect to x, of one of 
the auxiliary dependent variables. The equations (145), (146), and 
(147) form a normal system of equations of the first order. The 
initial conditions which must be satisfied by the integrals of this 
new system result immediately from the initial conditions imposed 
upon the integrals of the original system, and it is clear that the 
power series obtained for the integrals z,, z,,---, z, of the new sys- 
tem will be identical with the power series obtained for the integrals 
of the given system. These series are therefore ee (see § 25) 
in the neighborhood of the point (a,, a,, +--+, @, 


For example, the equation of the second order r=f (a, y, Z, p, q, 8, t) can be 
replaced by a system of three equations of the first order in the normal form, 


0z op op oa) our op 
ae —_ = Hy, zy oy BS 
pres a, =f ( »Y; 2 P,Q, — aaeey Bao 


If it is required that z= ¢(y), 02/dx = y(y), for x = 2p, the integrals of the 
auxiliary system must reduce respectively, for «= 2, to the functions ¢(y), 


¥(y), o(y)- 


This general theorem does not furnish a reply to all the questions 
which can be proposed on the existence of integrals of any system 
whatever of partial differential equations, for it applies only to sys- 
tems in the normal form considered. The most general systems have 
been the subject of a great number of studies, the most recent of 
which, due to Tresse, Riquier, and Delassus, have led to the gen- 
eral solution of the following problem: Given a system of m partial 
differential equations of any order in any number of independent 
and any number of dependent variables, to determine whether this 
system has any integrals and, if it has, to define the arbitrary quan- 
tities (constants or functions) upon which the integrals depend.* 

To sum up, every partial differential equation of any order in 
which both sides are analytic functions of their arguments has-an 
infinite number of analytic integrals, but we cannot say, in general, 
as in the ease of ordinary differential equations (§ 26), that all the 


* The investigations of Riquier have been collected by him in his work Sur les 
systemes d’équations aux dérivées partielles (1910). : 


V, Exs.] EXERCISES 28-7 


integrals are analytic functions of the independent variables. We 
have seen above (p. 255, ftn.) that it is not true for an equation 
of the first order. It is, moreover, easy to see this by elementary 
examples such as the equation p = 0, whose general integral is any 
function of y. 

The methods of the calculus of limits do not apply to the non- 
analytic equations. Let us consider, for example, the equation 


(148) p+af@, y)=9 


where J(#, y) is a continuous non-analytic function satisfying the 
Lipschitz condition with respect to y. We have proved in §§ 27-30 
that the differential ae: 


‘7 t= f(a, Y) 


“i 


has an infinite number of integrals which depend upon an arbitrary 
constant C. In order to conclude from this, as in § 31, the existence 
of an integral of the equation (148), it would be necessary to prove 
that all these integrals are defined by an equation of the form 
(x, y)=C, where the function ¢@ possesses continuous derivatives 
of the first order. We shall return to this question in the next 
volume. 


EXERCISES 
1. Integrate the partial differential equations 


axtp + (x4z + ary — ax?y?)q = 2 ax®yz — za?y3, 


2 2 x 
Clad ke Yo cee 
(x —6y)p + (102 — y)q = 6y? — 42? — 86 ay. 


2. Find the general equation of the surfaces which cut at right angles the 
spheres represented by the equation 


xr? + y2? + 2274+ 2az=0, 
where a is a variable parameter. 


Deduce from the result obtained some systems of three families of orthogonal 
surfaces. 


3. It is required to find the partial differential equation of the surfaces 
described by a straight line which moves so that it always meets a fixed straight 
line at a given angle. Integrate this partial differential equation. 

[ Licence, Paris, July, 18738.] 


4. Given a plane P and a poin\ O in the plane, find the general equation of 
all the surfaces such that, if we draw the normal mn at any point m of one of 
them meeting the plane P at n, and then the perpendicular mp to this plane, 
the area of the triangle Onp will be equal to a given constant. 

[ Licence, Paris, November, 1871.] 


288 PARTIAL DIFFERENTIAL EQUATIONS [V, Exs. 


5. The same question as in Ex. 4, supposing, however, that the angle nOp is 
constant. : 
[Licence, Rennes, 1883. ] 


6. Determine all the surfaces which satisfy the condition 
Op x mn = Om’, 


where \ denotes a given constant, O the origin of codrdinates, m any point of 
one of the surfaces, p the foot of the perpendicular dropped from O upon the 
tangent plane at m, and n the trace of the normal on the plane xOy. 

[Licence, Paris, 1875.] 


7. Find the general equation of the surfaces such that if we draw the 
normal mn from any point m of one of them terminating in the zy-plane, the 


length mn will be equal to the distance On. j eats * 
[ Licence, Poitiers, 1883. ] 
8. Find the integral surfaces of the equation 


cy*p + wyg =z (x? + y?). 
Determine the arbitrary function in such a way that the characteristic curves 
form a family of asymptotic lines of the integral surfaces, and find the 


orthogonal trajectories of the surfaces thus obtained. 
[ Licence, Paris, July, 1904. ] 


9. Consider a family of skew curves I represented by the two equations 
xe? + 2y? = az’, x? + y? + 22 = bg, . 


where a and 6 are two variable parameters, 

1) Prove that these curves are the orthogonal trajectories of a one-parameter 
family of surfaces S ; 

2) Find the lines of curvature of these surfaces S ; 

3) Show that these surfaces form part of a triply orthogonal system, and find 
the other two families of this system. ; 

[ Licence, Paris, July, 1901.] 

10. Form the partial differential equation which has the complete integral 

y? (x? — a) = (2 + b)?, and integrate this equation. 


11. Determine the surfaces such that the segment mn of the normal included 
between the surface and the point of intersection n with a fixed plane P pro- 
jects upon this plane into a segment of constant length. 


12. Let n be the point where the normal at m to a surface meets the zy- 
plane. Find the surfaces such that the straight line On will be parallel to the 


tangent plane at m. : [oe 
[ Licence, Poitiers, July, 1884.] 


13. It is required to determine the surfaces which cut at a given angle V all 
the planes passing through a fixed straight line. Show that the characteristic 
curves are the lines of curvature of the integral surfaces. 


14, The integral curves of the partial differential equation for which a com- 
plete integral is (1—a)2+k(1 + a2)z+2ay4+b=0, 
where a and 0 are two arbitrary constants, satisfy the relation 


dx? + dy? = k?dz?. 


V, Exs. ] EXERCISES 289 


15*, Every integral curve of a partial differential equation F(z, y, z, p,q) = 0, 
tangent at a point M to a generator G of the cone 7 with its vertex at M, 
has contact of the second order with every integral surface tangent at M to the 


plane tangent along the generator G to the cone T. 
[Sopnuus Lie.] 


16. From a point M of a surface S a perpendicular MP is dropped upon the 
fixed axis OO’, then from the point P a perpendicular PN upon the normal to 
the surface at M. It is required to determine the surfaces S such that the 
length MW will be a given constant a. 

Study in particular the surfaces S which are helicoids having OO’ for axis. 

[Licence, Paris, October, 1908.] 


17. It is required to find the general form of the functions F(a, y, z, p, q) 
such that the differential equations of the characteristic curves of the equation 
F = 0 will have the integrable combination d(q¢/p) = 0. 


Application. Determine the surfaces S such that the distance of any point 
M of one of them to the zy-plane is equal to the distance from the point O to 
the tangent plane to the surface at the point M. 


18. Given the partial differential equation 
(I) : Pp + Qq = Rz? + Sz + T, 


where P, Q, R, S, T depend only upon the variables z and y, show that the 
anharmonic ratio u of any four particular integrals of the equation (I) satisfies 
the equation 


Knowing four particular integrals z,, 2., 23, 2, of the equation (I), can we 
derive from them the general integral ? 


19. Parallel surfaces. Let 6 (a, y, z) be an integral of the equation 


: 2) + CY (Of 


Prove that the equation 6(z, y, 2) = C represents, in rectangular coérdinates, 
a family of parallel surfaces. 


Note. We observe that the equation (E) has the complete integral 
= V(a — a)? + (y — 6)? + (2-0), 


and the general integral is obtained by finding the envelope of the sphere of 
radius @ whose center describes a surface or a curve. It is clear that by 
making the radius 6 vary we obtain a family of parallel surfaces. 

Conversely, in order that the equation u(x, y,z) = C shall represent a family 
of parallel surfaces, it is necessary and sufficient (Ex. 9, p. 42) that u(az, y, 2) 
satisfy an equation of the form 


(28), 28)" 5 (2) 90, 


which we may reduce to the form (E) by putting @ = y(u). 


290 PARTIAL DIFFERENTIAL EQUATIONS [V, Exs. 


20. In order that the expression dz + Adz + Bdy shall have an integrating 
factor independent of z, it is necessary and sufficient that it be of the form 


dz + zd¢ + e-¢dy, 
where ¢ and y are functions of x and y. 
21. Apply the method of J. Bertrand (p. 232) to the equation 
Pdz+Qdy + Rdz=0, 


where P, Q, R are linear functions of a, y, z satisfying the condition of 
integrability. 
22*. Given a completely integrable system of the form 

dz = pdx + qdy, 

dp = (a, p + 29 + a,2) dx + (cp + Cog + C2) dy, 

dg = (p+ 6.9 + Cy Z) dx + (b,p + b,g + 6,2) dy, 
where a;, 0;, c; are functions of 2 and y, the general integral is of the form 
z= 0,2, + C.z, + C32,, where z,, 2, 2, are three linearly independent inte- 
grals, and where C,, C,, C; are arbitrary constants.* 


23. Find the necessary and sufficient conditions in order that the equations 
emg dee Y); $= 5 (a, Y), t=, (@, y) 
be consistent. 


Application. Find what condition the functions A(z, y), B(x, y), C(x, y) 
must satisfy in order that the integral curves of the differential equation 
A dz? + 2 Bdrdy + Cdy?=0 be the projections on the zy-plane of the two 
families of asymptotic lines of a surface. 


* APPELL, Journal de Liouville, 3d series, Vol. VIII, p. 192. 


INDEX 


[Titles in italic are proper names; numbers in italic are page numbers; and num- 
bers in roman type are paragraph numbers. | 


Abel: 28, 15 

Abelian integral: 18, 11 

Abel’s theorem, applied to Euler’s 
equation: 28, 15 

Adjoint equation, polynomial: 115, 
42; 116, 42 

Adjoint system of equations: 156, 57; 
166, 62 

Algebraic critical points: 173, 63; 
doe Ors 199, 415 LOL, tt 

Algebraic differential equations: 180, 
66; 182, 67 

Analytic extension of integrals: 101, 
87; 152, 56; 180, 66 ; 182, 67 ; 188, 68 

Analytic integrals: 45, 22; 49, 22; 
50, ftn.; 51, 28; 53, 24; 54, 25; 
57,25 ; 59, 26 ; 60, 26 ; 61,26 ; 67, 29; 
100, 87; 175, 64; 246,84; 284, 94 

Anharmonic ratio of integrals: 13, 7 

Antomari: 246, ftn. 

Appell: 41, ftn.; 115, 42; 290, ftn. 

Approximate integration of differen- 
tial equations: 64, ftn. 

Arbitrary constants: 74, 81; see also 
Elimination of constants 

Asymptotic lines: 43, ex.18; 91, 35; 
206, ex. 6 

Auxiliary equation, polynomial: 12, 6 ; 
117,43; 124,45; 163,60; roots of: 
119, 43; for a system of equations: 
158, 58; 163, 60 


Bernoulli: 11, 5 

Bernoulli’s equation: 11, 5 

Bertrand: 41, ftn.; 232, 80; 290, ex. 21 

Bertrand’s method: 232, 80; 290, ex. 
21 

Bessel: 126, 46; 142, 52; 169, ex. 8 

Bessel’s equation: 126, 46; 142, 52; 
169, ex. 8 

Boole: 212, ex. 1 

Bounitzky: 44, ex. 21 

Bracket [u, v]: 234, 81; 241, 83 

9 


aed 


91 


Briot and Bouquet: 45, 21; 50, ftn.; 
59, 26; 173, 64; 175, 64; 176, 64; 
Le 6, GA 175, 66's 19dfith, 

Briot and Bouquet’s equation : 173, 64 

Briot and Bouquet’s method, analytic 
integrals: 50, ftn.; 59, 26 

Briot and Bouquet’s theorem : 175, 64; 
176, 64; 177, 64; 178, 65 


Calculus of limits: 45, 21 and 22; 65, 
ftn. ; 137, 50; (system of equations) : 
48,22; equations of the nth order: 49, 
22; 100, 37; non-linear equations: 
174, 64; partial differential equa- 
tions: 53, 25; (system of): 56, 25; 
2838, 94; 287, 943; system of linear 
equations: 50, 23; total differential 
equation: 51,24; (system of): 53, 24 

Canonical form, of substitutions: 131, 
48; 132,48; of a system of linear 
equations : 161, 59 ; 165,61; 179, 65 

Cauchy: 35,18; 45, 21; 46, 22; 61, 
27; 68, 30; 73, 30; 74, 80; 108, 
89; 109, ftn.; 128, 46; 154, ftn.; 
172) 08185, 61} LOSS Tee Z0z eel’: 
214,75; 217,75; 246, 84; 249,85; 
254, 85; 257, 85; 257, Note; 259, 
86; 260, Note; 261, 87; 264, 87 

Cauchy-Lipschitz method : 61,27; 68, 
30; 74, 30 

Cauchy’s equation: 257, ex. 1. 

Cauchy’s first proof: 68, 30; 73, 30 

Cauchy’s method: non-homogeneous 
linear equations: 108, 39 ; 109, ftn.; 
(system of): 154, ftn.; partial dif- 
ferential equations: 249, 85; 257, 
Note; 259, 86; 260, Note; (ex- 
tended) : 261, 87 

Cauchy’s problem: 246, 84; 264, 87 

Cauchy’s theorem: 45, 22; 172, 63; 
188, 67; 198, 71; 202, 715 (system 
of equations) : 48, 22 ; 217, 75; par- 
tial differential equations: 54, 26 


292 


Center, of integral curves: 180, 65; 
of similitude, 8, 3 

Characteristic curves: 219, 76; 224, 
77; 249, 85; 250, 85; 259, 86; 
261, 87; Cauchy’s method : 249, 85; 
257, Note; 259, 86; 260, Note; 
261, 873; congruence of: 219, 76; 
220, 76; 222, 77; derivation from 
complete integral: 259, 86 ; differen- 
tial equations of: 219, 76; 222,77; 
224, 717; 251, 85; see also Char- 
acteristic strip 

Characteristic developable surface: 
252, 85; 259, 86; 260, Note 

Characteristic direction: 250, 85 

Characteristic equation: 130,47 ; 139, 
50; 140,51; 143,53; 147,54; 166, 
61; 166, 62; 179, 65; elementary 
divisors: 132, ftn.; roots of: 130, 
47; 131, 48; 189, 60; 149, Note 2 

Characteristic exponents: 147, 54; 
150, 55 

Characteristic numbers: 147, 54 

Characteristic strip: 252,85; 259, 86 ; 
260, Note; 261, 87; differential 
equation of : 262, 87 

Circles, differential equation of: 5,1; 
of double contact with a conic: 
206, ex. 4 

Cissoid: 206, ex. 5 

Clairaut: 17,10; 41, ftn.; 44, ex. 20; 
205, 72; 212, 74; 239, ex. 1 

Clairaut’s equation: 17,10; 47, ftn.; 
44, ex. 20; 205, 725; generalized: 
212, 74; 239, ex. 1 

Clebsch: 267, 88 

Complete integral: 236, 82; 239, 82; 
241, 83; 247, 84; 260, Note; 277, 
91; 278, 92; generalization of the- 
ory: 272,90; geometric interpreta- 
tion: 288, 82; of involutory systems: 
277, 91; see also Cauchy’s method 
and Lagrange’s theory 

Complete systems: 267, 88 and 89; 
equivalent: 268, 89; Jacobian sys- 
tems: 269, 89; 270, 89; 271, ex.; 
275, 91; 278, 92; method of inte- 
gration: 270, 89; change of varia- 
bles: 267, 89 


INDEX 


Completely integrable total differen- 
tial equations: 52, 24; 225, 78; 
system of equations: 53, 24; see 
also Condition for integrability 

Complex of curves: 259, 86 

Condition for incompressibility of a 
fluid: 86, 38 

Condition for integrability of total 
differential equations: 52, 24; 226, 
78; 230, 80; the bracket [u, v]: 
284, 81; 241, 83; invariance of: 
281, 80; involutory systems, Pois- 
son’s parenthesis: 274, 91; the 
parenthesis (u, v): 234, 81 

Conformal representation: 22, 18 

Congruence of curves: 209, 74; 219, 
76; 222, 773; focal points of, focal 
surface: 209, 74; 224, 77; see also 
Characteristic curves and Edge of 
regression 

Conical point: 257, 85 

Conics, differential equation of: 5,1; 
having circles of double contact: 
206, ex. 4 

Conoids: 220, ex. 1 

Constant coefficients in differential 
equations: 117,43; (system of equa- 
tions): 157, 58; 160, 58; D’Alem- 
bert’s method: 122, 44; 161, 58 

Constants of integration: 74, 31; see 
also Elimination of constants 

Continuous one-parameter groups: 
87, 34; see also Groups 

Corresponding homogeneous linear 
equation: 107, 39 

Cotton: 64, ftn. 

Covariant: 80, Note 2 

Cremona transformation: 198, ftn. 

Critical points, algebraic: 173, 63; 
183, 67; 199, 71; 201, 713 infinite 
number of: 185, ftn.; linear equa- 
tions: 129, 47; non-linear equa- 
tions: 178, 63; permutation of 
integrals about: 129, 47; 133, 49; 
transcendental: 197, 70 

Curves, asymptotic lines: 43, ex. 18; 
91, 85; 206, ex. 6; circles: see Cir- 
cles; cissoid: 206, ex. 5; complex 
of: 259, 86; congruence of: see 


INDEX 


Congruence of curves; conics: see 
Conics; cycloid: 47, 20; edge of 
regression: 209, 74; 212, 74; 240, 
ex. 2; 257, 853 elastic space curve: 
99, ex. 73 ellipse: 18, 103; enve- 
lope: see Envelope; family of: 3,1; 
helices: 220, ex. 2; isothermal: 43, 
ex. 125 orthogonal: 14, 7; 33,17; 
220, ex. 8; 228, 783; parabola: 6, 
1; parallel: 42, ex. 93 similar: 8, 
8; straight lines: 4,1; trajectories: 
14, 7; 34,17; 98, 36; unicursal 
quartic: 19, ex. 2; 205, 72; see also 
Cusps Integral curves, Lines of 
Curvature, Locus 

Cusps of integral curves: 41, 20; 
201, 71; 202,71; 208, 73; 212, 74; 
218, ex. 2; see also Locus of cusps 


Darboux: 29, 16; 41, fin.; 45, 21; 
79, ttn. s "116 ttn, 3 205,.ftn. 3, 223, 
ex. 2; 239, ftn.; 253, 85 

Darboux’s theorems: 29, 16 

D’ Alembert: 122, 44; 161, 58 

D’ Alembert’s method: 122, 44; 161,58 

Definite integrals as solutions, of Bes- 
sel’s equation: 126,46; 169, ex. 8; 
of Laplace’s equation: 124, 46 

Delassus: 286, 94 

Depression of order: 36, 19; 109, 40 

Derivative in non-linear equations, 
infinite: 172, 63; indeterminate: 
178, 64; 177, 65; see also Briot and 
Bouquet’s equation and Briot and 
Bouquet’s theorem 

Developable surfaces: 240, 82; 257, 
85; 282, ex.; see also Character- 
istic developable surfaces 

Differential equations: 3, 13; admit- 
ting a group of transformations: 
89,35; 91,353 95,36; 96; 36; 97, 
36; 98, ex. 435 algebraic: 180, 66; 
182, 67; algebraic, of deficiency 
zero or one: 18, 11; Bernoulli’s: ZZ, 
5; Bessel’s: 126,46; 142,52; 169, 
ex. 8; Briot and Bouquet’s: 173, 64; 
Cauchy’s: 257, ex. 1; of character- 
istic curves: 219, 76; 222, 77; 224, 
77; 251, 85; of characteristic strip: 


293 


262, 873 of circles: 5,1; Clairaut’s: 
see Clairaut’s equation; of conics 
(Halphen’s method): 5, 13; Dar- 
boux’s theorems: 29, 16; depression 
of order of: 36,19; 109, 40; differ- 
ential notation: 7, 2; elastic space 
curve: 99, ex. 7; equations F (a, y’) 
= 0, Miy, 7 p= Os 18 113° Euler's: 
see Euler’s equation; Euler’s linear: 
123, 45; existence theorems: see 
Existence theorems; of first order: 
6, 2; 180, 663; Gauss’s: 140, 51; 
geometric representation of: 14, 8; 
of higher order: 35, 18; 196, 70; 
homogeneous: 8,3; 16, ftn.; 38,19; 
90, 85; of incompressible fluid: 84, 
33; integrals of: see Integral curves, 
Integral surfaces, and Integrals; of 
isothermal curves: 43, ex. 12; Jaco- 
bi’s: see Jacobi’s equation; La- 
grange’s: 16,9; 204, 72; 205, 72; 
-Lamé’s: 146, 53; Laplace’s linear: 
124, 46; Legendre’s: 112, ex.; lin- 
ear: 9,4; 90, 35; Liouville’s: 79, 
ex. 8; of the nth order: 4,1; 6, 
2; 49, 22; 100, 387; order of: 4, 
1; of orthogonal trajectories: 14, 
Ve88, V1 228,- 185 Painlevé’s: 
196, 70; 197, 70; of parabolas: 6,, 
1; with periodic coefficients: see 
Periodic coefficients; Picard’s: 143, 
58; raising order of: 41, Note; 
regular: 134, 50; Riccati’s: see Ric- 
cati’s equation; of similar curves: 
&, 3; singular points of : see Singu- 
lar points; of straight lines: 4, 1; 
of trajectories: see Trajectories; see 
also special classes of differential 
‘equations and systems of equations 
Differential notation: 7, 2 
Differential operators: 97, 36; 102, 
88; 113, 413; bracket [u, v]: 234, 
81; 241, 83; the parenthesis (u, v): 
234, 81; Poisson’s parenthesis: 274, 
91; X[Y(f)]—Y[A(S)]: 97, 86; 
266, 88; 278, 92 
Dixon: 44, ex. 21 
Dominant functions: 45, 21; 47, 22; 51, 
28: 52,24; 55,25; 188, 50; 174, 64 


294 


Doubly periodic functions of the sec- 
ond kind: 144, 53 


Edge of regression: 209, 74; 212, 74; 
240, ex. 2; 257, 85 | 

Elastic space curve: 99, ex. 7 

Element: 251, 85; 261, 87 

Elementary divisors: 132, ftn. 

Elimination, of arbitrary functions: 
222,77; 288,82; 259,86; 278, 90; 
278, 93; of constants: 3, 1; 208, 
74; 286, 82; 255, ftn.; 272, 90 

Ellipsoid, lines of curvature of: 42, 
Note 

Elliptic functions: 23, 143 as coeffi- 
cients of a linear equation: 144, 
58; 146, 54; existence proof from 
Euler’s equation: 23, 14; 194, 69; 
as integrals: 19, ex. 3; 39,20; 144, 
538; 192, 68; Picard’s equation: 
- 144, 53 

Envelope, of asymptotic lines: 206, 
ex. 6; of integral curves: 77, 10; 
208, 71; 204, ftn. ; 205, 72; 209, 74; 
218, ex. 8; of integral surfaces: 288, 
82 ; 281, 93; of straight lines: 18, 10 

Equations of first order, higher order: 
see Differential equations and special 
classes of equations 

Equivalent complete systems: 268, 89 

Essentially singular points: 131, 47; 
134, 49; movable: 196, 70 

Kuler’: 19, 12:;° 23,145 27,.145>28, 
16'5°29, 0163.41, fing: 43, Ox Sea. 
117, 43; 123,45; 194, 69; 205, 72; 
221, ex. 8 

Euler’s equation: 23,14; 28,15; 41, 
ftn.; 194, 69; 205, 72; Abel’s the- 
orem: 28, 15; existence of elliptic 
functions: 28, 14; 194, 69; La- 
grange’s integral of: 43, ex. 17; 
Stieltjes’s general integral: 27, 14 

Euler’s linear equation: 123, 45 

Euler’s relation for homogeneity: 221, 
ex. 3 

Exceptional initial values: 172, 63; 
173, 64; 177, 65 

Existence theorems: 45, 22; 98, ex. 
1; analytic integrals: see Analytic 


INDEX 


integrals and Briot and Bouquet’s 
method; calculus of limits: see 
Calculus of limits; for elliptic func- 
tions: 23, 14; 194, 69; for inte- 
grating factors: 57, 263 successive 
approximations: see Successive ap- 
proximations; for systems of partial 
differential equations in normal 
form: 283, 94; see also Exceptional 
initial values 
Extended group: 94, 36 


First integrals: 74, 381; 76, 81; 981, 
32; 83, 82; 157, 57; 216, 75 

Fixed singular points : 181, 66 ; 182, 67 

Floquet: 151, ftn. 

Focal point: 209, 74 

Focal surface: 209, 74; 224, 77 

Focus: 180, 65 ; 

Fuchs :-184, 50; 139; ftns; 160, ftng: 
194, ftn. 

Fuchs’ theorem: 134, 50 

Functions defined by differential 
equations: 182, 67 

Fundamental characteristic equation : 
189, 50; see also Characteristic 
equation 

Fundamental system of integrals: 
108, 38; 105, 38; 129,47; 130, 47; 
147, 54; for a system of linear 
equations: 1538, 56 


Gauss: 140, 51 

Gauss’s equation: 140, 51 

General integral: 3,1; 12, 7; 59, 26; 
74, 81; of homogeneous linear 
equations: 103, 88; 105, 38; of 
partial differential equations: 217, 
75; 288, 82; 2738, 90; of a system 
of equations: 57, 26; 152, 56 

Goursat: 83, ftn.; 170, exs. 14, 15; 
208, ftn.; 265, ftn. 

Group, differential equations admit- 
ting a: 89, 385; 91, 35; 95, 36; 96, 
36; 97, 86; 98, ex. 4; differential 
equations of a: 88, 34 

Groups, one-parameter continuous: 
86, 34; 91, 36; application to differ- 
ential equations: 89, 35; 96, 36; 


INDEX 


97,36; functions admitting: 93, 36; 
of infinitesimal transformations: 
91, 36; 93, 86; invariants: 93, 36; 
similar: 88, 34; of translations: 
89, 84; see also Transformations 


Halphen: 5,1; 115, 42 

Hedrick: 255, ftn. 

Helices: 220, ex. 2; 245, 83 

Helicoid: 220, ex.2; 245, 83; lines 
of curvature of: 91, 35 

Hermite: 99, ex. 7; 146, 53; 169, ex. 
85293; ftn.: 283, ftn: 

Homogeneity of functions, Euler’s re- 
lation: 221, ex. 3 

Homogeneous equations: 8, 3; 6, 
ftn.; 38,19; 90, 35 

Homogeneous linear equations: 102, 
38; 107, 39; adjoint equation, poly- 
nomial: 116, 42; analogies with 
algebraic equations: 713, 415 anal- 
ogies with the Galois theory, with 
symmetric functions of roots: 115, 
41; auxiliary equation, polynomial: 
117, 433 Bessel’s equation: 126, 46; 
142, 52; 169, ex. 8; common inte- 
grals of two equations: 114, 41; 
constant coefficients: 117, 43; 
(D’Alembert’s method): 122, 44; 
corresponding: 107, 893 critical 
points: 729, 47; depression of order: 
109, 403 elliptic coefficients: 144, 
53; 146,54; Euler’s linear equation: 
123, 45; Fuchs’ theorem: 134, 50; 
fundamental system of integrals: 
103, 38; 105, 88; 129, 473; Gauss’s 
equation : 140,51; general integral: 
103, 38; 105, 883; greatest common 
divisor: 113, 41; group of substitu- 
tions: 132, 48; 134, 48; invariants: 
115, 41; Lamé’s equation: 146, 58; 
Laplace’s equation : 124, 46; Legen- 
dre’s equation: 112, ex.; linearly 
independent integrals: 103, 38; 105, 
38; periodic coefficients: 128, 47; 
146, 54; 150, ex.; 151, ftn.; per- 
mutations of integrals around a 
critical point: 129, 473; Picard’s 
equation: 143, 533 ratio of two 


295 


integrals: 169, ex. 10; regular: 134, 
50; regular integrals: 128, 47; 131, 
47; 134, 49; relation to Riccati’s 
equation: 111, 40; 112, ftn.; roots 
of integrals, Sturm’s theorem: 1/7, 
ftn.; solution as a definite inte- 
gral: see Definite integrals; system 
of: see System of homogeneous lin- 
ear equations; Wronskian: 129, 47; 
see also Characteristic equation, 
Characteristic numbers, and Char- 
acteristic exponents 

Hotiel: 212, ex. 1 

Hyperelliptic functions: 193, 68 

Hypergeometric series: 140, 513; de- 
generate cases: 142, 52 


Identical transformation: 88, 34; 91, 
36 

Incompressible fluid, condition for: 
8&6, 83; invariant integrals: 8&4, 33 

Independent equations: 265, 88 

Independent integrals: 872, 315 line- 
arly: 103, 38; 105, 38 

Infinitesimal transformations: 86, 34 ; 
91, 36; 93, 36; 98, 36 

Initial conditions: 45, 22; 48, 22; 49, 
22; 50, 23; 52, 24; 53, 24; 61, 26; 
defining an integral: 100, 37; partial 
differential equations: 54, 25; 57, 
25; 214,74; 221,76; 246,84; 284, 
94; see also Cauchy’s problem, De- 
rivatives in non-linear equations, 
and Exceptional initial values 

Integrable combination: 77, 81; 78, 
exs. 1,2; 220, 76; 245, 88 

Integral curves: 4, 1; 60, 26 and ftn.; 
61, 26; 79, Note 1; 173, ftn.; 179, 
65; 299,713 center: 780, 655 cusps: 
see Cusps; envelope of: 27,10; 203, 
71; 204, ftn.; 205, 72; 209, 74; 213, 
ex. 8; focus: 180, 65; in para- 
metric form: 16,93; of a partial dif- 
ferential equation: 257, 85; 258, 
ex. 2; 289, ex. 15; saddleback: 
179, 65; see also Integrals 

Integral equation: 61, 27 

Integral surfaces: 218, 76; 219, 76; 
227,78; 246, 84; 250,85; 255, 85; 


296 


envelope of: 238, 82; 281, 98; see 
also Cauchy’s problem and Integrals 
Integrals: Abelian: 18,115; analytic: 
see Analytic extension and Analytic 
integrals; anharmonic ratio of: 13, 
7; Cauchy’s problem: 246, 84; 264, 
87; complete: see Complete inte- 
gral; common to two linear equa- 
tions: 114, 41; defined by initial 
conditions: 100, 87; in form of 
definite integrals: see Definite in- 
tegrals; elements of: 251, 85; 261, 
87; elliptic functions: 19, ex. 3; 39, 
20; 144, 58; 192, 68; of equations 
of higher order: 196, 703 existence 
of: see Existence theorems; first: 
see First integrals; fundamental 
system of: see Fundamental system 
of integrals; general: see General 
integral; general properties of: 
100, 37; hypergeometric series: 140, 
51; 142, 52; independent: 81, 31; 
initial conditions: see Initial condi- 
tions; invariant: see Invariant inte- 
grals; Legendre’s polynomials: 112, 
ex.; Lie’s enlarged definition: 264, 
Note; linearly independent: 103, 
88; 105, 38; non-analytic: see Non- 
analytic integrals; particular: 3,1; 
12, 7;.14, 7; 20,12; 107, 39; 109, 
40; periodic: 192, 68; permutation 
of integrals around a critical point: 
129,47; 133,49; rational functions: 
144,58; 192, 68; rational functions 
of constants: 10,4; 12, 7; 186, 67; 
regular: 128, 47; 131, 47; 134, 49; 
roots of, Sturm’s theorem: 1/71, 
ftn.; singular: see Singular inte- 
erals; singular points: see Singular 
points; Wronskian: 129, 47; see 
also Integrable combination, Inte- 
gral curves, Integral surfaces, and 
special types of equations 
Integrating factors: 19, 12; 81,32; 83, 
32; 96,36; 98, exs. 3,4; 115, 42; 231, 
80 ; 290, ex. 203 existence of : 57, 26 
Integration by raising order: 41, Note 
Invariance of condition ofintegrability: 
231, 80 


INDEX 


Invariant functions: 93, 36 

Invariant integral: 83, 33; of homoge- 
neous linear equations: 115, 41; of 
incompressible fluid: 84, 33; line 
and surface: 84, 33; multiple: 85, 
835 volume: 86, 33 

Involutory systems: 274, 91; com- 
plete integral: 277, 91; Jacobi’s 
method: 277, 92; Poisson’s paren- 
thesis: 274, 91 

Isothermal curves: 43, ex. 12 


Jacobi: 11,6; 25,14; 32,16; 74, 31; 
81, 82; 163, 60; 269, 89; 270, 89; 
271, ex.; 275, 91; 277, 92; 278, 92 

Jacobi’s equation: 11, 6; 32, 163 re- 
lation to a system of homogeneous 
linear equations: 163, 60 

Jacobi’s method, involutory systems: 
277, 92 

Jacobi’s multipliers: 74, 31; 81, 82 

Jacobian system: 269, 89; 270, 89; 
27 1, OX. 39278, 91 5, 27 S702 


Kovalevsky, Madame: 45, 21; 2838, 94 
and ftn. 


Lagrange: 16,9; 41, ftn.; 43,ex.17; 
107, 39; 109, ftn.; 115, 42; 208, 
71; 204, 712; 205, 72; 2138, ex. 4; 
236, 82; 239, 82; 240, 88; 241, 83; 
251, 85; 255, ftn.; 258, ex. 1; 259, 

86; 277, 92 

Lagrange and Charpit’s method : 240, 
83; 277, 92 

Lagrange’s equation: 16,9; 204, 72; 
205, 72 

Lagrange’s integral of Euler’s equa- 
tion: 43, ex. 17 

Lagrange’s method: 241, 83; 251, 85 

Lagrange’s method of the variation of 
constants: 107, 39; 109, ftn.; 255, 
ftn. 

Lagrange’s theory of the complete in- 
tegral: 236, 82; 289, 82; 258, ex. 
1; 259, 86 

Laguerre: 115, 41 

Lamé: 146, 58 

Lamé’s equation: 146, 53 


INDEX 


Laplace: 124, 46; 127, Note 

Laplace’s linear equation: 124, 46; 
127, Note 

Legendre: 16, ftn.; 28,15; 112, ex. 

Legendre’s equation: 112, ex. 

Legendre’s polynomials: 172, ex. 

Legendre’s transformation: 16, ftn. 

Leibnitz : 118, 48 

Leibnitz’s formula: 118, 48 

Liapunof: 151, 55 and ftn.; 166, 62 

Lie: 438, ex. 12; 86, ftn.; 98, 36; 
264, Note; 289, ex. 15 

Lie’s enlarged definition of the inte- 
gral: 264, Note 

Lie’s theory of differential equations: 
86, 34; see also Groups 

Lindelof: 61, 27; 98, ex. 1 

Linear equations: 9,4; 90, 385; 100, 
37; 106, 39; 186, 673; coefficients 
depending upon a parameter: 65, 
Note ; depression of order : 109, 40; 
general properties of integrals: 100, 
37; see also Homogeneous linear 
equations, Integrals, Non-homoge- 
neous linear equations, Partial differ- 
ential equations, and Singular points 

Linearly independent functions: 103, 
38; integrals: 103, 38; 105, 38 

Lines of curvature: 206, 72; of an 
ellipsoid : 41, Note; of a helicoid: 
91, 35 

Liouville: 79, ex. 3 

Liouville’s equation: 79, ex. 8 

Lipschitz : 68, 80; 287, 94 

Lipschitz condition : 68, 30; 287, 94 

Locus, of characteristic curves: 219, 
76; of cusps of integral curves: 
201, 71; 202, 71; 206, 72; 208, 73; 
212, 74; 213, ex. 23 of points of 
‘inflection of integral curves: 223, 
ex. 2, 


Mayer: 229, 79 

Mayer’s method: 229, 79 

Méray: 45, 21 

Moigno: 68, 30; 212, ex. 1 

Monge: 41, Note 

Monge’s method of finding the lines of 
curvature of an ellipsoid: 42, Note 


297 


Movable singular points: 181, 66; 
185, 67; 197, 70 and ftn.; for 
equations of higher order: 196,70; 
essentially singular: 196, 70; lines 
of : 197,70; poles: 197, ftn.; tran- 
scendental critical points: 197, 70 

Multipliers: 74, 81; 81, 32; 85, 33 


Non-analytic integrals: 50, 22; 255, 
ftn.; Briot and Bouquet’s theorem : 
175, 64; 176, 64; 177, 64; 178, 65; 
see also Analytic integrals and Briot 
and Bouquet’s method 

Non-homogeneous linear equations: 
100, 37; 106, 393; analytic exten- 
sion of integrals: 101,37; Cauchy’s 
method: 108, 39; 109, ftn.; con- 
stant coefficients: 120, 48; corre- 
sponding homogeneous equation : 
107, 39; depression of order: 110, 
40; general integral: 107, 39; La- 
grange’s method of the variation of 
constants: 107,39; 109, ftn.; singular 
points: 100,87 ; system of equations: 
see Systems of non-homogeneous 
linear equations 

Non-linear differential equations, 172, 
63; 179, 65; algebraic equations of 
the first order: 180, 66; 182, 673 
Briot and Bouquet’s problem: 193, 
ftn.; having single-valued inte- 
grals: 187, 68; 192, 685, 193, ftn.3 
exceptional initial values: 172, 6335 
(derivative infinite): 172, 635 (de- 
rivative indeterminate): 173, 64; 
177, 65; integrals: see Envelope of 
integrals, Integral curves, Integrals, 
Locus of cusps, and Singular inte- 
erals; functions defined by y’ = 
R(x, y): 182,67; non-analytic inte- 
grals: see Non-analytic integrals; 
singular points: see Critical points, 
Fixed singular points, Movable sin- 
gular points, Singular points; sys- 
tems of: 208, 74; see also Equations 
of Briot and Bouquet, Clairaut, 
Euler, Lagrange, and Riccati 

Normal form of a system of partial 
differential equations: 283, 94 


s 


298 


Order of a differential equation: 4,1; 
depression of : 36,19 ; 109, 40; first 
order: 6,2; 180, 66; higher order, 
mth order: 4,1; 6,2; 35, 18; 196, 
70; integration by raising order: 
41, Note 

Orthogonal trajectories: 14, 7; 33, 
17; 220, ex. 3; 228, 78 

Orthogonal surfaces: 223, 77 


Painlevé: 59, ftn.; 74,30; 196, 70; 
197,70 ; 218, ex. 7 

Painlevé’s equation: 196, 70; 
70 

Parallel curves: 42, ex. 9 

Parallel surfaces: 289, ex. 19 

Parenthesis (u, v): 234, 81; Poisson’s: 
274, 91 

Partial differential equations: 76, 313 
of first order: see Partial differential 
equations of the first order; of 
higher order: 278, 935; (system of 
equations): 283, 94; of ruled sur- 
faces: 280, ex. 1; 281, ex. 25 of 
tubular surfaces: 240, ex. 3; 282, 
ex. ; see also Systems of differential 
equations and Existence theorems 

Partial differential equations of the 
first order, linear: 75, 31; 214, 75; 
characteristic curves: see Character- 
istic curves; of conoids: 220, ex. 1; 
general integral: 217, 75; geomet- 
ric interpretation: 218, 76; general 
method of integration: 214, 75; of 
helicoids: 220, ex. 2; 245, 83; initial 
conditions, 221, 76; integral surface: 
218, 76; 219, 76; singular integral, 
surface: 224, 77; see also Systems 
of differential equations 

Partial differential equations of the 
first order, non-linear: any num- 
ber of variables: 261, 87; Cauchy’s 
equation: 257, ex. 1; Cauchy’s 
method: 249, 85; 259, 86; 260, 
Note; (extended): 261,87; Cauchy’s 
problem: 246, 84; characteristic 
curves, characteristic developable 
surface, characteristic direction, 
characteristic strip: see these titles; 


197, 


INDEX 


Clairaut’s equation, generalized: 
289, ex. 1; complete integral: 236, 
82, and see also Lagrange’s theory $5 
element: 251, 85; envelope of sur- 
faces: 238, 82; general integral: 
288, 82; integral, Lie’s enlarged 
definition: 264, Note; integral 
curves: 257, 85; 289, ex. 15; La- 
grange and Charpit’s method: 240, 
83; 277,92; separation of variables: 
244, ex. 3; singular integrals: 224, 
17; 287, 82; 238, itn.; 272, 00; 
three variables: 236, 82; see also 
Involutory systems 

Particular integral, solution: 3, 1; 
42,13 I8,ets 20s 1236 ee 
109, 40 

Periodic coefficients: 128, 47; 146, 
54; 150, ex.; 151, ftn.$ elliptic: 
144, 58; 146, 54; Picard’s equa- 
tion: 144, 53; system of linear 
equations: 164, 61; 166, 62 

Picard: 59, ftn.; 61, 27; 74,30; 115, 
42; 144,53; 177, 64 

Picard’s equation: 144, 53 

Picard’s method of successive approxi- 
mations : see Successive approxima- 
tions 

Poincaré: 83,83; 126, {tn.; 151, ftn.; 
177, 64; 180, 65; 194, ftn. 

Poisson: 274, 91; 277, 92 

Poisson’s identity: 277, 92 

Poisson’s parenthesis: 274, 91 

Poles of integrals: 143, 58; 183, 67; 
185,67; movable: 186,67; 197,ftn. 

Properties, of differential equations of 
higher order: 196, 70; of e*, tanz: 
218, ex. 6 


Quadratures: 7,2; 10,4; 12,7; 18, 
13.14, 73° 16,9 29, 12% 7S ole eee 
Note 1; 83, 82; 90, 35; 108, 39; 
110, 40; 111,40; 154,56; 278, 92 


Raffy: 44, ex. 21 

Ratio, of similitude: 8, 3; of two 
integrals: 169, ex. 10 

Rational functions, of constants as in- 
tegrals: 10, 4; 12, 7; 186, 67; of 


INDEX 


variables as integrals: 144, 58; 192, 
68 

Reducible systems: 165, 62 

Regular differential equations: 134, 
50 

Regular integrals: 128, 47; 131, 47; 
184,49; Fuchs’ theorem: 134, 50; 
substitutions: 132, 48; 134, 48 

Tceatis 12h. 13,.1tn:7, 48, ex. 18; 
79, ex. 2; 111,40; 112, 40 and ftn.; 
143, Note; 157,57; 169,ex.9; 170, 
ex. 15; 186, 67; 187, 67 and ftn.; 
pods. 197, ftn.3) 215, 6x, 7 

Riccati’s equation: 12, 7; 43, ex. 13; 
79, ex.2; 111,40; 112, 40 and ftn.; 
143, Note; 157,57; 169,ex.9; 170, 
ex. 15;,181, 67 and ftn.; 186,-67 ; 
194, ftn.; 213, ex. 7; generalization 
of : 197, ftn.; linear transformation 
of: 13, ftn.; relation to linear equa- 
tions: 711, 40; 112, ftn. 

Riquier: 45, 21; 286, 94 and ftn. 

Roots of characteristic equation: 130, 
47; 131, 48; 139, 50; 149, Note 2; 
elementary divisors: 132, ftn. 

Roots of integrals: Sturm’s theorem: 
111, ftn. 

Ruled surfaces: 280, ex.1; 281, ex. 2 


Saddleback: 179, 65 

Sauvage: 182, ftn. 

Schlomilch : 212, ex. 1 

Separation of variables: 6, 2; 8, 3; 
19,12; 244, ex. 3 

Serret.: 212, ex. 1; 213, ex. 3 

Similar curves: 8, 3 

Similar groups: 88, 34 

Single-valued integrals: 144, 58; of 
(y’)™ = R (y), classification of equa- 
tions: 187, 68; 192, 68; 198, ftn. 

Singular integral, curve, surface: 17, 
LOst 27, 145 76, ftn.; 198, 71+ 202, 
71; 205, 72 ; 206,72; 208, 74; 210, 
14; 224,77; 237, 82;;, 255, ftn.3.as 
an envelope: 203,71; 238, 82; geo- 
metric interpretation : 207; 78 

Singular integral, curves and surfaces: 
determination of: 205,72; of first- 
order equations: 198, 71; 202,71; 


299 


206, 72; geometric interpretation : 
207, 733; of partial differential equa- 
tions: 224, 77; 287, 82; 2388, ftn. ; 
272, 90; of a system of equations: 
208, 74; 210, 74 

Singular lines, movable: 197, 70 

Singular points: algebraic critical 
points: 173, 63; 183, 67; 184, 67; 
201, 71; Briot and Bouquet’s theo- 
rem: 176, 64; center: 180, 65; of 
equations of the first order: 180, 
66; essentially : 131, 47; 134, 49; 
essentially singular movable: 196, 
70; fixed: 181,66; 182, 67; focus: 
180, 65; of linear equations: 65, 
28; 100,87; 129,47; 140,51; 142, 
52; 143, 53; indeterminate deriva- 
tive: 173, 64; infinite derivative : 
172, 63; infinite number of critical 
points: 185, ftn.; movable: see 
Movable singular points; poles: 131, 
47; 143, 53; 144, 58; 188, 67; 184, 
67; 185,67; 197, {tn.; saddleback : 
179, 65 

Solution : see Integral 

Star: 67, 29 

Stationary flow: 86, 33 

Stieltjes: 27, 14 

Stieltjes’s general integral of Euler’s 
equation: 27, 14 

Straight lines, differential equation 
of: 4, 1 

Sturm: 111, ftn. 

Sturm’s theorem: 1717, ftn. 

Substitutions: linear equations: 129, 
47; 1382, 48; 134, 48; canonical 
form: 131, 48 ; 132, 48; system of 
linear equations, canonical form : 
165, 61; Wronskian: 129, 47 

Successive approximations: 61, 27; 
analytic functions: 66,29; 102, 37; 
175, 64; Cauchy-Lipschitz method : 
61, 27; 68, 30; 74, 30; Cauchy’s first 
proof: 68, 30; 73, 30; coefficients 
functions of a parameter: 65, Note; 
Lindeléf’s addition: 67, 27; linear 
equations: 64, 28; Lipschitz condi- 
tion: 68, 30; 287, 94 ; non-analytic 
integrals: 175, 64; real variables: 


300 


61, 27; 62, 27; 68,80; 73, 30; 150, 
553 star: 67, 29 

Surfaces, conoids: 220, ex. 1; develop- 
able: 240, 82; 257, 85; 282, ex.; 
ellipsoid: 41, Note; focal: 209, 
74; 224, 77; helicoids: 220, ex. 2; 
245, 835 orthogonal: 223, 773; par- 
allel: 289, ex. 193 ruled: 280, ex. 
1; 281, ex. 2; tubular: 240, ex. 3; 
282, ex.; see also Characteristic de- 
velopable surfaces, Envelopes, and 
Integral surfaces 

Symbolic polynomial: 113, 41; 116, 
42; 118,433 divisor: 114, 41; great- 
est common divisor: 113, 41 

Systems of differential equations: 60, 
26; 74, 81; 79, Note 15 covariant: 
80, Note 2; existence theorem: 
see Existence theorem; first inte- 
grals: see First integrals; general 
integral: 57, 263; integral curve: 
60, 26; invariant integrals: see 
Invariant integrals; multipliers: 
74, 31; 81, 82; 85, 33; singular inte- 
grals: 208, 74; see also Integrable 
combination, Systems of homoge- 
neous linear equations, and Systems 
of non-homogeneous linear equations 

Systems of homogeneous linear equa- 
tions: 152, 563; adjoint system: 


156, 57; 166, 623; auxiliary equa- . 


tion: 158, 583; canonical form: 
161, 59; 165, 61; 179, 653; con- 
stant coefficients: 157, 58; 160, 
58; (D’Alembert’s method): 62, 
58; fundamental system of inte- 
grals: 153, 563 periodic coefficients : 
164,61; 166, 62; reducible systems: 
165, 623; relation to Jacobi’s equa- 
tion: 163, 60; substitutions: 165, 61 

Systems of non-homogeneous linear 
equations: 154, 56; Cauchy’s 
method: 154, ftn.; existence theo- 
rem: 50, 23 

Systems of partial differential equa- 
tions: of first order: 272, 903; nor- 
mal form, general existence theo- 
rem: 283, 94; see also Existence 
theorems, Involutory systems, and 


INDEX 


Systems of homogeneous linear par- 
tial differential equations of the first 
order 

Systems of partial differential equa- 
tions, homogeneous linear equa- 
tions of the first order: 265, 88; 
independent equations: 265, 88; 
X[Y(/)]-Y¥[X(f)]: 266, 88; see 
also Complete systems 


Tannery: 139, ftn. 

Taylor: 35, 18 

Total differential equations: 51, 24; 
225, 78; 241, 83; 276, 913; Ber- 
trand’s method: 232, 80; 290, ex. 21; 
completely integrable: 52, 24; 225, 
78; existence theorem: 451, 24; 
geometric interpretation: 227, 78; 
integral surface: 227, 78; Mayer’s 
method: 229, 79; method of inte- 
gration: 225, 78; 232, 80; Pdzx + 
Qdy + Rdz=0: 230, 80; see also 
Condition of integrability 

Trajectories: 13, 7; 14, 7; 34, 17; 
93, 36 , 

Transcendental critical points: 197,70 

Transformations: 82, 82; 83, 32; 
84,33; admitting a group of: 89,35; 
96, 86 5; of complete systems : 267,89 5 
covariants: 80, Note 2; Cremona: 
198, ftn.; extended group of: 94,36; 
identical: 88, 34; 91, 863 infinites- 
imal: 86, 34; 91,36; 93, 36; 98,36; 
inverse: 89, 84; Legendre’s: 6, 
ftn.; of linear equations: 115, 41; 
162, 59; of Riccati’s equation: 73, 
ftn. ; see also Groups and Invariants 

Tresse: 286, 94 

Tubular surfaces: 240, ex.3; 282, ex. 


Unicursal quartic: 19, ex. 2; 205, 72 


Variation of constants: 107, 39; 109, 
ftn.; 255, ftn. 


Weierstrass: 45, 21; 132, 48 and ftn. 

Weierstrass’s elementary divisors: 
132, ftn. 

Wronskian: 129, 47 


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